An infinite dimensional umbral calculus
Dmitri Finkelshtein, Yuri Kondratiev, Eugene Lytvynov, Maria Joao Oliveira
AAn infinite dimensional umbral calculus
Dmitri Finkelshtein
Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP,U.K.; e-mail: [email protected]
Yuri Kondratiev
Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, 33615 Bielefeld, Germany;e-mail: [email protected]
Eugene Lytvynov
Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP,U.K.; e-mail: [email protected]
Maria Jo˜ao Oliveira
Departamento de Ciˆencias e Tecnologia, Universidade Aberta, 1269-001 Lisbon, Por-tugal; CMAF-CIO, University of Lisbon, 1749-016 Lisbon, Portugal;e-mail: [email protected]
Abstract
The aim of this paper is to develop foundations of umbral calculus on the space D (cid:48) of distri-butions on R d , which leads to a general theory of Sheffer polynomial sequences on D (cid:48) . Wedefine a sequence of monic polynomials on D (cid:48) , a polynomial sequence of binomial type, anda Sheffer sequence. We present equivalent conditions for a sequence of monic polynomialson D (cid:48) to be of binomial type or a Sheffer sequence, respectively. We also construct a lift-ing of a sequence of monic polynomials on R of binomial type to a polynomial sequence ofbinomial type on D (cid:48) , and a lifting of a Sheffer sequence on R to a Sheffer sequence on D (cid:48) .Examples of lifted polynomial sequences include the falling and rising factorials on D (cid:48) , Abel,Hermite, Charlier, and Laguerre polynomials on D (cid:48) . Some of these polynomials have alreadyappeared in different branches of infinite dimensional (stochastic) analysis and played therea fundamental role. Keywords:
Generating function; polynomial sequence on D (cid:48) ; polynomial sequenceof binomial type on D (cid:48) ; Sheffer sequence on D (cid:48) ; shift-invariance; umbral calculus on D (cid:48) . Secondary:
In its modern form, umbral calculus is a study of shift-invariant linear operators actingon polynomials, their associated polynomial sequences of binomial type, and Sheffersequences (including Appell sequences). We refer to the seminal papers [29, 38, 39],1 a r X i v : . [ m a t h . F A ] M a r ee also the monographs [24, 37]. Umbral calculus has applications in combinatorics,theory of special functions, approximation theory, probability and statistics, topology,and physics, see e.g. the survey paper [12] for a long list of references.Many extensions of umbral calculus to the case of polynomials of several, or eveninfinitely many variables were discussed e.g. in [5,10,14,28,32,35,36,41,42], for a longerlist of such papers see the introduction to [13]. Appell and Sheffer sequences of poly-nomials of several noncommutative variables arising in the context of free probability,Boolean probability, and conditionally free probability were discussed in [2–4], see alsothe references therein.The paper [13] was a pioneering (and seemingly unique) work in which elementsof basis-free umbral calculus were developed on an infinite dimensional space, moreprecisely, on a real separable Hilbert space H . This paper discussed, in particular, shift-invariant linear operators acting on the space of polynomials on H , Appell sequences,and examples of polynomial sequences of binomial type.In fact, examples of Sheffer sequences, i.e., polynomial sequences with generatingfunction of a certain exponential type, have appeared in infinite dimensional analysis onnumerous occasions. Some of these polynomial sequences are orthogonal with respectto a given probability measure on an infinite dimensional space, while others are relatedto analytical structures on such spaces. Typically, these polynomials are either definedon a co-nuclear space Φ (cid:48) (i.e, the dual of a nuclear space Φ), or on an appropriatesubset of Φ (cid:48) . Furthermore, in majority of examples, the nuclear space Φ consists of(smooth) functions on an underlying space X . For simplicity, we choose to work inthis paper with the Gel’fand tripleΦ = D ⊂ L ( R d , dx ) ⊂ D (cid:48) = Φ (cid:48) . Here D is the nuclear space of smooth compactly supported functions on R d , d ∈ N ,and D (cid:48) is the dual space of D , where the dual pairing between D (cid:48) and D is obtainedby continuously extending the inner product in L ( R d , dx ).Let us mention several known examples of Sheffer sequences on D (cid:48) or its subsets:(i) In infinite dimensional Gaussian analysis, also called white noise analysis, Her-mite polynomial sequences on D (cid:48) (or rather on S (cid:48) ⊂ D (cid:48) , the Schwartz space oftempered distributions) appear as polynomials orthogonal with respect to Gaus-sian white noise measure, see e.g. [6, 15, 16, 31].(ii) Charlier polynomial sequences on the configuration space of counting Radon mea-sures on R d , Γ ⊂ D (cid:48) , appear as polynomials orthogonal with respect to Poissonpoint process on R d , see [17, 19, 22].(iii) Laguerre polynomial sequences on the cone of discrete Radon measures on R d , K ⊂ D (cid:48) , appear as polynomials orthogonal with respect to the gamma randommeasure, see [21, 22]. 2iv) Meixner polynomial sequences on D (cid:48) appear as polynomials orthogonal with re-spect to the Meixner white noise measure, see [25, 26].(v) Special polynomials on the configuration space Γ ⊂ D (cid:48) are used to constructthe K -transform, see e.g. [7, 18, 20]. Recall that the K -transform determines theduality between point processes on R d and their correlation measures. Thesepolynomials will be identified in this paper as the infinite dimensional analog ofthe falling factorials (a special case of the Newton polynomials).(vi) Polynomial sequences on D (cid:48) with generating function of a certain exponential typeare used in biorthogonal analysis related to general measures on D (cid:48) , see [1, 23].Note, however, that even the very notion of a general polynomial sequence on aninfinite dimensional space has never been discussed!The classical umbral calculus on the real line gives a general theory of Sheffersequences and related (umbral) operators. So our aim in this paper is to developfoundations of umbral calculus on the space D (cid:48) , which will eventually lead to a generaltheory of Sheffer sequences on D (cid:48) and their umbral operators. In fact, at a structurallevel, many results of this paper will have remarkable similarities to the classical settingof polynomials on R . For example, the form of the generating function of a Sheffersequence on D (cid:48) will be similar to the generating function of a Sheffer sequence on R :the constants appearing in the latter function are replaced in the former function byappropriate linear continuous operators.There is a principal point in our approach that we would like to stress. The paper[13] deals with polynomials on a general Hilbert space H , while the monograph [6]develops Gaussian analysis on a general co-nuclear space Φ (cid:48) , without the assumptionthat Φ (cid:48) consists of generalized functions on R d (or on a general underlying space).In fact, we will discuss in Remark 7.5 below that the case of the infinite dimensionalHermite polynomials is, in a sense, exceptional and does not require from the co-nuclearspace Φ (cid:48) any special structure. In all other cases, the choice Φ (cid:48) = D (cid:48) is crucial. Havingsaid this, let us note that our ansatz can still be applied to a rather general co-nuclearspace of generalized functions over a topological space X , equipped with a referencemeasure.We also stress that the topological aspects of the spaces D , D (cid:48) , and their sym-metric tensor powers are crucial for us since all the linear operators (appearing as‘coefficients’ in our theory) are continuous in the respective topologies. Furthermore,this assumption of continuity is principal and cannot be omitted.The origins of the classical umbral calculus are in combinatorics. So, by analogy, onecan think of umbral calculus on D (cid:48) as a kind of spatial combinatorics. To give the readera better feeling of this, let us consider the following example. Let γ = (cid:80) ∞ i =1 δ x i ∈ Γbe a configuration. Here δ x i denotes the Dirac measure with mass at x i . We willconstruct (the kernel of) the falling factorial, denoted by ( γ ) n , as a function from Γ to3 (cid:48)(cid:12) n . (Here and below (cid:12) denotes the symmetric tensor product.) This will allow usto define ‘ γ choose n ’ by (cid:0) γn (cid:1) := n ! ( γ ) n . And we will get the following explicit formulawhich supports this term: (cid:18) γn (cid:19) = (cid:88) { i ,...,i n }⊂ N δ x i (cid:12) δ x i (cid:12) · · · (cid:12) δ x in , (1.1)i.e., the sum is obtained by choosing all possible n -point subsets from the (locally finite)set { x i } i ∈ N . The latter set can be obviously identified with the configuration γ .The paper is organized as follows. In Section 2 we discuss preliminaries. In partic-ular, we recall the construction of a general Gel’fand triple Φ ⊂ H ⊂ Φ (cid:48) , where Φ isa nuclear space and Φ (cid:48) is the dual of Φ (a co-nuclear space) with respect to the centerHilbert space H . We consider the space P (Φ (cid:48) ) of polynomials on Φ (cid:48) and equip it with anuclear space topology. Its dual space, denoted by F (Φ (cid:48) ), has a natural (commutative)algebraic structure with respect to the symmetric tensor product. We also define afamily of shift operators, ( E ( ζ )) ζ ∈ Φ (cid:48) , and the space of shift-invariant continuous linearoperators on P (Φ (cid:48) ), denoted by S ( P (Φ (cid:48) )).In Section 3, we give the definitions of a polynomial sequence on Φ (cid:48) , a monicpolynomial sequence on Φ (cid:48) , and a monic polynomial sequence on Φ (cid:48) of binomial type.Starting from Section 4, we choose the Gel’fand triple as D ⊂ L ( R d , dx ) ⊂ D (cid:48) . Themain result of this section is Theorem 4.1, which gives three equivalent conditions for amonic polynomial sequence to be of binomial type. The first equivalent condition is thatthe corresponding lowering operators are shift-invariant. The second condition gives arepresentation of each lowering operator through directional derivatives in directionsof delta functions, δ x ( x ∈ R d ). The third condition gives the form of the generatingfunction of a polynomial sequence of binomial type.To prove Theorem 4.1, we derive two essential results. The first one is an operatorexpansion theorem (Theorem 4.7), which gives a description of any shift-invariantoperator T ∈ S ( P ( D (cid:48) )) in terms of the lowering operators in directions δ x . The secondresult is an isomorphism theorem (Theorem 4.9): we construct a bijection J betweenthe spaces S ( P ( D (cid:48) )) and F ( D (cid:48) ) such that, under J , the product of any shift-invariantoperators goes over into the symmetric tensor product of their images. This implies,in particular, that any two shift-invariant operators commute.Next, we define a family of delta operators on D (cid:48) and prove that, for each suchfamily, there exists a unique monic polynomial sequence of binomial type for whichthese delta operators are the lowering operators.In Section 5, we identify a procedure of the lifting of a polynomial sequence ofbinomial type on R to a polynomial sequence of binomial type on D (cid:48) . This becomespossible due to the structural similarities between the one-dimensional and infinite-dimensional theories. Using this procedure, we identify, on D (cid:48) , the falling factorials,the rising factorials, the Abel polynomials, and the Laguerre polynomials of binomial4ype. We stress that the polynomial sequences lifted from R to D (cid:48) form a subset of a(much larger) set of all polynomial sequences of binomial type on D (cid:48) .In Section 6, we define a Sheffer sequence on D (cid:48) as a monic polynomial sequenceon D (cid:48) whose lowering operators are delta operators. Thus, to every Sheffer sequence,there corresponds a (unique) polynomial sequence of binomial type. In particular, if thecorresponding binomial sequence is just the set of monomials (i.e., their delta operatorsare differential operators), we call such a Sheffer sequence an Appell sequence. Themain result of this section, Theorem 6.2, gives several equivalent conditions for a monicpolynomial sequence to be a Sheffer sequence. In particular, we find the generatingfunction of a Sheffer sequence on D (cid:48) .In Section 7, we extend the procedure of the lifting described in Section 5 to Sheffersequences. Thus, for each Sheffer sequence on R , we define a Sheffer sequence on D (cid:48) .Using this procedure, we recover, in particular, the Hermite polynomials, the Charlierpolynomials, and the orthogonal Laguerre polynomials on D (cid:48) .Finally, in Appendix, we discuss several properties of formal tensor power series.From the technical point of view, the similarities between the infinite dimensionaland the classical settings open new perspectives in infinite dimensional analysis. Due tothe special character of this approach, namely, through definition of umbral operatorson P ( D (cid:48) ) and umbral composition of polynomials on D (cid:48) , further developments andapplications in infinite dimensional analysis are subject of forthcoming publications.Let us also mention the open problem of (at least partial) characterization of Sheffersequences that are orthogonal with respect to a certain probability measure on D (cid:48) . Inthe one-dimensional case, such a characterization is due to Meixner [27]. For multi-dimensional extensions of this result, see [11, 33, 34] and the references therein. Let us first recall the definition of a nuclear space, for details see e.g. [8, Chapter 14,Section 2.2]. Consider a family of real separable Hilbert spaces ( H τ ) τ ∈ T , where T isan arbitrary indexing set. Assume that the set Φ := (cid:84) τ ∈ T H τ is dense in each Hilbertspace H τ and the family ( H τ ) τ ∈ T is directed by embedding, i.e., for any τ , τ ∈ T there exists a τ ∈ T such that H τ ⊂ H τ and H τ ⊂ H τ and both embeddings arecontinuous. We introduce in Φ the projective limit topology of the H τ spaces:Φ = proj lim τ ∈ T H τ . By definition, the sets { ϕ ∈ Φ | (cid:107) ϕ − ψ (cid:107) H τ < ε } with ψ ∈ Φ, τ ∈ T , and ε > (cid:107) · (cid:107) H τ denotes the norm in H τ .5ssume that, for each τ ∈ T , there exists a τ ∈ T such that H τ ⊂ H τ , and theoperator of embedding of H τ into H τ is of the Hilbert–Schmidt class. Then the lineartopological space Φ is called nuclear .Next, let us assume that, for some τ ∈ T , each Hilbert space H τ with τ ∈ T iscontinuously embedded into H := H τ . We will call H the center space .Let Φ (cid:48) denote the dual space of Φ with respect to the center space H , i.e., the dualpairing between Φ (cid:48) and Φ is obtained by continuously extending the inner product in H , see e.g. [8, Chapter 14, Section 2.3]. The space Φ (cid:48) is often called co-nuclear .By the Schwartz theorem (e.g. [8, Chapter 14, Theorem 2.1]), Φ (cid:48) = (cid:83) τ ∈ T H − τ ,where H − τ denotes the dual space of H τ with respect to the center space H . Weendow Φ (cid:48) with the Mackey topology—the strongest topology in Φ (cid:48) consistent with theduality between Φ and Φ (cid:48) (i.e., the set of continuous linear functionals on Φ (cid:48) coincideswith Φ). The Mackey topology in Φ (cid:48) coincides with the topology of the inductive limitof the H − τ spaces, see e.g. [40, Chapter IV, Proposition 4.4] or [6, Chapter 1, Section1]. Thus, we obtain the Gel’fand triple (also called the standard triple )Φ = proj lim τ ∈ T H τ ⊂ H ⊂ ind lim τ ∈ T H − τ = Φ (cid:48) . (2.1)Let X and Y be linear topological spaces that are locally convex and Hausdorff.(Both Φ and Φ (cid:48) are such spaces.) We denote by L ( X, Y ) the space of continuous linearoperators acting from X into Y . We will also denote L ( X ) := L ( X, X ). We denote by X (cid:48) and Y (cid:48) the dual space of X and Y , respectively. We endow X (cid:48) with the Mackeytopology with respect to the duality between X and X (cid:48) . We similarly endow Y (cid:48) withthe Mackey topology.Each operator A ∈ L ( X, Y ) has the adjoint operator A ∗ ∈ L ( Y (cid:48) , X (cid:48) ) (also calledthe transpose of A or the dual of A ), see e.g. [30, Theorem 8.11.3]. Remark . Note that, since we chose the Mackey topology on Φ (cid:48) , for an operator A ∈ L (Φ (cid:48) ), we have A ∗ ∈ L (Φ). This fact will be used throughout the paper. Proposition 2.2.
Consider the Gel’fand triple (2.1) . Let A : Φ → Φ and B : Φ (cid:48) → Φ (cid:48) be linear operators. (i) We have A ∈ L (Φ) if and only if, for each τ ∈ T , there exists a τ ∈ T suchthat the operator A can be extended by continuity to an operator ˆ A ∈ L ( H τ , H τ ) . (ii) We have B ∈ L (Φ (cid:48) ) if and only if, for each τ ∈ T , there exists a τ ∈ T suchthat the operator ˆ B := B (cid:22) H − τ takes on values in H − τ and ˆ B ∈ L ( H − τ , H − τ ) .Remark . Proposition 2.2 admits a sraightforward generalization to the case of twoGel’fand triples, Φ ⊂ H ⊂ Φ (cid:48) and Ψ ⊂ G ⊂ Ψ (cid:48) , and linear operators A : Φ → Ψ and B : Φ (cid:48) → Ψ (cid:48) . Remark . Part (ii) of Proposition 2.2 is related to the universal property of aninductive limit, which states that any linear operator from an inductive limit of a6amily of locally convex spaces to another locally convex space is continuous if andonly if the restriction of the operator to any member of the family is continuous, seee.g. [9, II.29].
Proof of Proposition 2.2. (i) By the definition of the topology in Φ, the linear operator A : Φ → Φ is continuous if and only if, for any τ ∈ T and ε >
0, there exist τ ∈ T and ε > { ϕ ∈ Φ | (cid:107) ϕ (cid:107) H τ < ε } contains the set { ϕ ∈ Φ | (cid:107) ϕ (cid:107) H τ < ε } . But this implies the statement.(ii) Assume B ∈ L (Φ (cid:48) ). Then, by Remark 2.1, we have B ∗ ∈ L (Φ). Hence, for each τ ∈ T , there exists a τ ∈ T such that the operator B ∗ can be extended by continuityto an operator ˆ B ∗ ∈ L ( H τ , H τ ). But the adjoint of the operator ˆ B ∗ is ˆ B := B (cid:22) H − τ .Hence ˆ B ∈ L ( H − τ , H − τ ).Conversely, assume that, for each τ ∈ T , there exists a τ ∈ T such that theoperator ˆ B := B (cid:22) H − τ takes on values in H − τ and ˆ B ∈ L ( H − τ , H − τ ). Therefore,ˆ B ∗ ∈ L ( H τ , H τ ). Denote A := ˆ B ∗ (cid:22) Φ. As easily seen, the definition of the operator A does not depend on the choice of τ , τ ∈ T . Hence, A : Φ → Φ, and by part (i) weconclude that A ∈ L (Φ). But B = A ∗ and hence B ∈ L (Φ (cid:48) ).In what follows, ⊗ will denote the tensor product. In particular, for a real separableHilbert space H , H ⊗ n denotes the n th tensor power of H . We will denote by Sym n ∈L ( H ⊗ n ) the symmetrization operator, i.e., the orthogonal projection satisfyingSym n f ⊗ f ⊗ · · · ⊗ f n = 1 n ! (cid:88) σ ∈ S ( n ) f σ (1) ⊗ f σ (2) ⊗ · · · ⊗ f σ ( n ) (2.2)for f , f , . . . , f n ∈ H . Here S ( n ) denotes the symmetric group acting on { , . . . , n } .We will denote the symmetric tensor product by (cid:12) . In particular, f (cid:12) f (cid:12) · · · (cid:12) f n := Sym n f ⊗ f ⊗ · · · ⊗ f n , f , f , . . . , f n ∈ H , and H (cid:12) n := Sym n H ⊗ n is the n th symmetric tensor power of H . Note that, for each f ∈ H , we have f (cid:12) n = f ⊗ n .Starting with Gel’fand triple (2.1), one constructs its n th symmetric tensor poweras follows: Φ (cid:12) n := proj lim τ ∈ T H (cid:12) nτ ⊂ H (cid:12) n ⊂ ind lim τ ∈ T H (cid:12) n − τ =: Φ (cid:48)(cid:12) n , see e.g. [6, Section 2.1] for details. In particular, Φ (cid:12) n is a nuclear space and Φ (cid:48)(cid:12) n is itsdual with respect to the center space H (cid:12) n . We will also denote Φ (cid:12) = H (cid:12) = Φ (cid:48)(cid:12) := R .The dual pairing between F ( n ) ∈ Φ (cid:48)(cid:12) n and g ( n ) ∈ Φ (cid:12) n will be denoted by (cid:104) F ( n ) , g ( n ) (cid:105) .7 emark . Consider the set { ξ ⊗ n | ξ ∈ Φ } . By the polarization identity, the linearspan of this set is dense in every space H (cid:12) nτ , τ ∈ T .The following lemma will be very important for our considerations. Lemma 2.6. (i)
Let F ( n ) , G ( n ) ∈ Φ (cid:48)(cid:12) n be such that (cid:104) F ( n ) , ξ ⊗ n (cid:105) = (cid:104) G ( n ) , ξ ⊗ n (cid:105) for all ξ ∈ Φ , then F ( n ) = G ( n ) . (ii) Let Φ and Ψ be nuclear spaces and let A, B ∈ L (Φ (cid:12) n , Ψ) . Assume that Aξ ⊗ n = Bξ ⊗ n for all ξ ∈ Φ . Then A = B .Proof. Statement (i) follows from Remark 2.5, statement (ii) follows from Proposi-tion 2.2, (i) and Remarks 2.3 and 2.5.
Below we fix the Gel’fand triple (2.1).
Definition . A function P : Φ (cid:48) → R is called a polynomial on Φ (cid:48) if P ( ω ) = n (cid:88) k =0 (cid:104) ω ⊗ k , f ( k ) (cid:105) , ω ∈ Φ (cid:48) , (2.3)where f ( k ) ∈ Φ (cid:12) k , k = 0 , , . . . , n , n ∈ N := { , , , . . . } , and ω ⊗ := 1. If f ( n ) (cid:54) = 0,one says that the polynomial P is of degree n . We denote by P (Φ (cid:48) ) the set of allpolynomials on Φ (cid:48) . Remark . For each P ∈ P (Φ (cid:48) ), its representation in form (2.3) is evidently unique.For any f ( k ) ∈ Φ (cid:12) k and g ( n ) ∈ Φ (cid:12) n , k, n ∈ N , we have (cid:104) ω ⊗ k , f ( k ) (cid:105)(cid:104) ω ⊗ n , g ( n ) (cid:105) = (cid:104) ω ⊗ ( k + n ) , f ( k ) (cid:12) g ( n ) (cid:105) , ω ∈ Φ (cid:48) . (2.4)Hence P (Φ (cid:48) ) is an algebra under point-wise multiplication of polynomials on Φ (cid:48) .We will now define a topology on P (Φ (cid:48) ). Let F fin (Φ) denote the topological directsum of the nuclear spaces Φ (cid:12) n , n ∈ N . Hence, F fin (Φ) is a nuclear space, see e.g. [6,Section 5.1]. This space consists of all finite sequences f = ( f (0) , f (1) , . . . , f ( n ) , , , . . . ),where f ( k ) ∈ Φ (cid:12) k , k = 0 , , . . . , n , n ∈ N . The convergence in F fin (Φ) means theuniform finiteness of non-zero elements and the coordinate-wise convergence in eachΦ (cid:12) k . 8 emark . Below we will often identify f ( n ) ∈ Φ (cid:12) n with(0 , . . . , , f ( n ) , , , . . . ) ∈ F fin (Φ) . We define a natural bijective mapping I : F fin (Φ) → P (Φ (cid:48) ) by( If )( ω ) := n (cid:88) k =0 (cid:104) ω ⊗ k , f ( k ) (cid:105) , (2.5)for f = ( f (0) , f (1) , . . . , f ( n ) , , , . . . ) ∈ F fin (Φ). We define a nuclear space topology on P (Φ (cid:48) ) as the image of the topology on F fin (Φ) under the mapping I .The space F fin (Φ) may be endowed with the structure of an algebra with respectto the symmetric tensor product f (cid:12) g = (cid:32) k (cid:88) i =0 f ( i ) (cid:12) g ( k − i ) (cid:33) ∞ k =0 , (2.6)where f = ( f ( k ) ) ∞ k =0 , g = ( g ( k ) ) ∞ k =0 ∈ F fin (Φ). The unit element of this (commutative)algebra is the vacuum vector Ω := (1 , , . . . ).By (2.4)–(2.6), the bijective mapping I provides an isomorphism between the alge-bras F fin (Φ) and P (Φ (cid:48) ), namely, for any f, g ∈ F fin (Φ), (cid:0) I ( f (cid:12) g ) (cid:1) ( ω ) = ( If )( ω )( Ig )( ω ) , ω ∈ Φ (cid:48) . Let F (Φ (cid:48) ) := ∞ (cid:89) k =0 Φ (cid:48)(cid:12) k denote the topological product of the spaces Φ (cid:48)(cid:12) k . The space F (Φ (cid:48) ) consists of allsequences F = ( F ( k ) ) ∞ k =0 , where F ( k ) ∈ Φ (cid:48)(cid:12) k , k ∈ N . Note that the convergence inthis space means the coordinate-wise convergence in each space Φ (cid:48)(cid:12) k .Each element F = ( F ( k ) ) ∞ k =0 ∈ F (Φ (cid:48) ) determines a continuous linear functional on F fin (Φ) by (cid:104) F, f (cid:105) := ∞ (cid:88) k =0 (cid:104) F ( k ) , f ( k ) (cid:105) , f = ( f ( k ) ) ∞ k =0 ∈ F fin (Φ) (2.7)(note that the sum in (2.7) is, in fact, finite). The dual of F fin (Φ) is equal to F (Φ (cid:48) ), andthe topology on F (Φ (cid:48) ) coincides with the Mackey topology on F (Φ (cid:48) ) that is consistentwith the duality between F fin (Φ) and F (Φ (cid:48) ), see e.g. [6]. In view of the definition ofthe topology on P (Φ (cid:48) ), we may also think of F (Φ (cid:48) ) as the dual space of P (Φ (cid:48) ).Similarly to (2.6), one can introduce the symmetric tensor product on F (Φ (cid:48) ): F (cid:12) G = (cid:32) k (cid:88) i =0 F ( i ) (cid:12) G ( k − i ) (cid:33) ∞ k =0 , (2.8)9here F = ( F ( k ) ) ∞ k =0 , G = ( G ( k ) ) ∞ k =0 ∈ F (Φ (cid:48) ). The unit element of this algebra is againΩ = (1 , , , . . . ).We will now discuss another realization of the space F (Φ (cid:48) ). We denote by S ( R , R )the vector space of formal series R ( t ) = (cid:80) ∞ n =0 r n t n in powers of t ∈ R , where r n ∈ R for n ∈ N . The S ( R , R ) is an algebra under the product of formal power series. Similarlyto S ( R , R ), we give the following Definition . Each ( F ( n ) ) ∞ n =0 ∈ F (Φ (cid:48) ) identifies a ‘real-valued’ formal series (cid:80) ∞ n =0 (cid:104) F ( n ) , ξ ⊗ n (cid:105) in tensor powers of ξ ∈ Φ. We denote by S (Φ , R ) the vector space ofsuch formal series with natural operations. We define a product on S (Φ , R ) by (cid:32) ∞ (cid:88) n =0 (cid:104) F ( n ) , ξ ⊗ n (cid:105) (cid:33) (cid:32) ∞ (cid:88) n =0 (cid:104) G ( n ) , ξ ⊗ n (cid:105) (cid:33) = ∞ (cid:88) n =0 (cid:42) n (cid:88) i =0 F ( i ) (cid:12) G ( n − i ) , ξ ⊗ n (cid:43) , (2.9)where ( F ( n ) ) ∞ n =0 , ( G ( n ) ) ∞ n =0 ∈ F (Φ (cid:48) ). Remark . Assume that, for some ( F ( n ) ) ∞ n =0 , ( G ( n ) ) ∞ n =0 ∈ F (Φ (cid:48) ) and ξ ∈ Φ, bothseries (cid:80) ∞ n =0 (cid:104) F ( n ) , ξ ⊗ n (cid:105) and (cid:80) ∞ n =0 (cid:104) G ( n ) , ξ ⊗ n (cid:105) converge absolutely. Then also the serieson the right hand side of (2.9) converges absolutely and (2.9) holds as an equality oftwo real numbers. Remark . Let t ∈ R and ξ ∈ Φ. Then, tξ ∈ Φ and for ( F ( n ) ) ∞ n =0 ∈ F (Φ (cid:48) ), ∞ (cid:88) n =0 (cid:104) F ( n ) , ( tξ ) ⊗ n (cid:105) = ∞ (cid:88) n =0 t n (cid:104) F ( n ) , ξ ⊗ n (cid:105) , (2.10)the expression on the right hand side of equality (2.10) being the formal power seriesin t that has coefficient (cid:104) F ( n ) , ξ ⊗ n (cid:105) by t n .According to the definition of S (Φ , R ), there exists a natural bijective mapping I : F (Φ (cid:48) ) → S (Φ , R ) given by( I F )( ξ ) := ∞ (cid:88) n =0 (cid:104) F ( n ) , ξ ⊗ n (cid:105) , F = ( F ( n ) ) ∞ n =0 ∈ F (Φ (cid:48) ) , ξ ∈ Φ . (2.11)The mapping I provides an isomorphism between the algebras F (Φ (cid:48) ) and S (Φ , R ),namely, for any F, G ∈ F (Φ (cid:48) ), (cid:0) I ( F (cid:12) G ) (cid:1) ( ξ ) = ( I F )( ξ )( I G )( ξ ) (2.12)see (2.8) and (2.9). Remark . In view of the isomorphism I , we may think of S (Φ , R ) as the dual spaceof P (Φ (cid:48) ). 10nalogously to Definition 2.10 and Remark 2.12, we can introduce a space of Φ-valued tensor power series. Definition . Let ( A n ) ∞ n =1 be a sequence of operators A n ∈ L (Φ (cid:12) n , Φ). Then theoperators ( A n ) ∞ n =1 can be identified with a ‘ Φ -valued’ formal series (cid:80) ∞ n =1 A n ξ ⊗ n intensor powers of ξ ∈ Φ. We denote by S (Φ , Φ) the vector space of such formal series.
Remark . Let t ∈ R and ξ ∈ Φ. Then, tξ ∈ Φ and for a sequence ( A n ) ∞ n =1 as inDefinition 2.14 ∞ (cid:88) n =1 A n ( tξ ) ⊗ n = ∞ (cid:88) n =1 t n A n ξ ⊗ n is the formal power series in t that has coefficient A n ξ ⊗ n ∈ Φ by t n . Recall that, byLemma 2.6, (ii), the values of the operator A n on the vectors ξ ⊗ n ∈ Φ (cid:12) n uniquelyidentify the operator A n .In Appendix, we discuss several properties of formal tensor power series. Definition . For each ζ ∈ Φ (cid:48) , we define the operator D ( ζ ) : P (Φ (cid:48) ) → P (Φ (cid:48) ) ofdifferentiation in direction ζ by( D ( ζ ) P )( ω ) := lim t → P ( ω + tζ ) − P ( ω ) t , P ∈ P (Φ (cid:48) ) , ω ∈ Φ (cid:48) . Definition . For each ζ ∈ Φ (cid:48) , we define the annihilation operator A ( ζ ) ∈ L ( F fin (Φ))by A ( ζ )Ω := 0 , A ( ζ ) ξ ⊗ n := n (cid:104) ζ, ξ (cid:105) ξ ⊗ ( n − for ξ ∈ Φ , n ∈ N . Lemma 2.18.
For each ζ ∈ Φ (cid:48) , we have D ( ζ ) ∈ L ( P (Φ (cid:48) )) .Proof. Using the bijection I : F fin (Φ) → P (Φ (cid:48) ) defined by (2.5), we easily obtain D ( ζ ) = I A ( ζ ) I − , ζ ∈ Φ (cid:48) , which implies the statement. Definition . For each ζ ∈ Φ (cid:48) , we define the operator E ( ζ ) : P (Φ (cid:48) ) → P (Φ (cid:48) ) of shiftby ζ by ( E ( ζ ) P )( ω ) := P ( ω + ζ ) , P ∈ P (Φ (cid:48) ) , ω ∈ Φ (cid:48) . Lemma 2.20. (Boole’s formula) For each ζ ∈ Φ (cid:48) , E ( ζ ) = ∞ (cid:88) k =0 k ! D ( ζ ) k . roof. Note that the infinite sum is, in fact, a finite sum when applied to a polynomial,and thus it is a well-defined operator on P (Φ (cid:48) ). For each ξ ∈ Φ and n ∈ N , (cid:0) E ( ζ ) (cid:104)· ⊗ n , ξ ⊗ n (cid:105) (cid:1) ( ω ) = (cid:104) ( ω + ζ ) ⊗ n , ξ ⊗ n (cid:105) = n (cid:88) k =0 (cid:18) nk (cid:19) (cid:104) ω, ξ (cid:105) k (cid:104) ζ, ξ (cid:105) n − k = n (cid:88) k =0 k ! (cid:0) D ( ζ ) k (cid:104)· ⊗ n , ξ ⊗ n (cid:105) (cid:1) ( ω ) = ∞ (cid:88) k =0 k ! (cid:0) D ( ζ ) k (cid:104)· ⊗ n , ξ ⊗ n (cid:105) (cid:1) ( ω ) , which implies the statement.Note that Lemmas 2.18 and 2.20 imply that E ( ζ ) ∈ L ( P (Φ (cid:48) )) for each ζ ∈ Φ (cid:48) . Definition . We say that an operator T ∈ L ( P (Φ (cid:48) )) is shift-invariant if T E ( ζ ) = E ( ζ ) T for all ζ ∈ Φ (cid:48) . We denote the linear space of all shift-invariant operators by S ( P (Φ (cid:48) )). The space S ( P (Φ (cid:48) )) is an algebra under the usual product (composition) of operators.Obviously, for each ζ ∈ Φ (cid:48) , the operators D ( ζ ) and E ( ζ ) belong to S ( P (Φ (cid:48) )). We will now introduce the notion of a polynomial sequence on Φ (cid:48) .For each n ∈ N , we denote by P ( n ) (Φ (cid:48) ) the subspace of P (Φ (cid:48) ) that consists of allpolynomials on Φ (cid:48) of degree ≤ n . Lemma 3.1.
A mapping P ( n ) : Φ (cid:12) n → P ( n ) (Φ (cid:48) ) is linear and continuous if and only ifit is of the form (cid:0) P ( n ) f ( n ) (cid:1) ( ω ) = (cid:104) P ( n ) ( ω ) , f ( n ) (cid:105) , (3.1) where P ( n ) : Φ (cid:48) → Φ (cid:48)(cid:12) n is a continuous mapping of the form P ( n ) ( ω ) = n (cid:88) k =0 U n,k ω ⊗ k (3.2) with U n,k ∈ L (Φ (cid:48)(cid:12) k , Φ (cid:48)(cid:12) n ) .Proof. Let P ( n ) ∈ L (cid:0) Φ (cid:12) n , P ( n ) (Φ (cid:48) ) (cid:1) . Then, by the definition of P ( n ) (Φ (cid:48) ), there existoperators V k,n ∈ L (Φ (cid:12) n , Φ (cid:12) k ), k = 0 , , . . . , n , such that, for any f ( n ) ∈ Φ (cid:12) n and ω ∈ Φ (cid:48) (cid:0) P ( n ) f ( n ) (cid:1) ( ω ) = n (cid:88) k =0 (cid:104) ω ⊗ k , V k,n f ( n ) (cid:105) n (cid:88) k =0 (cid:104) U n,k ω ⊗ k , f ( n ) (cid:105) , where U n,k := V ∗ k,n ∈ L (Φ (cid:48)(cid:12) k , Φ (cid:48)(cid:12) n ). Conversely, every P ( n ) of the form (3.2) determines P ( n ) ∈ L (cid:0) Φ (cid:12) n , P ( n ) (Φ (cid:48) ) (cid:1) by formula (3.1). Definition . Assume that, for each n ∈ N , P ( n ) : Φ (cid:48) → Φ (cid:48)(cid:12) n is of the form (3.2)with U n,k ∈ L (Φ (cid:48)(cid:12) k , Φ (cid:48)(cid:12) n ). Furthermore, assume that, for each n ∈ N , U n,n ∈ L (Φ (cid:48)(cid:12) n )is a homeomorphism. Then we call ( P ( n ) ) ∞ n =0 a polynomial sequence on Φ (cid:48) .If additionally, for each n ∈ N , U n,n = , the identity operator on Φ (cid:48)(cid:12) n , then wecall ( P ( n ) ) ∞ n =0 a monic polynomial sequence on Φ (cid:48) . Remark . Below, to simplify notations, we will only deal with monic polynomialsequences. The results of this paper can be extended to the case of a general polynomialsequence on Φ (cid:48) = D (cid:48) . Remark . By the definition of a monic polynomial sequence we get (cid:104) P ( n ) ( ω ) , f ( n ) (cid:105) = (cid:104) ω ⊗ n , f ( n ) (cid:105) + n − (cid:88) k =0 (cid:104) ω ⊗ k , V k,n f ( n ) (cid:105) , f ( n ) ∈ Φ (cid:12) n (3.3)where V k,n := U ∗ n,k ∈ L (Φ (cid:12) n , Φ (cid:12) k ). Lemma 3.5.
Let ( P ( n ) ) ∞ n =0 be a monic polynomial sequence on Φ (cid:48) . The followingstatements hold. (i) There exist operators R k,n ∈ L (Φ (cid:12) n , Φ (cid:12) k ) , k = 0 , , . . . , n − , n ∈ N , such that,for all ω ∈ Φ (cid:48) and f ( n ) ∈ Φ (cid:12) n , (cid:104) ω ⊗ n , f ( n ) (cid:105) = (cid:104) P ( n ) ( ω ) , f ( n ) (cid:105) + n − (cid:88) k =0 (cid:104) P ( k ) ( ω ) , R k,n f ( n ) (cid:105) . (3.4)(ii) We have P (Φ (cid:48) ) = (cid:40) n (cid:88) k =0 (cid:104) P ( k ) , f ( k ) (cid:105) | f ( k ) ∈ Φ (cid:12) k , k = 0 , , . . . , n, n ∈ N (cid:41) . (3.5) Proof. (i) We prove by induction on n . For n = 1, the statement trivially holds.Assume that the statement holds for 1 , , . . . , n . Then, by using (3.3) and the inductionassumption, we get (cid:104) ω ⊗ ( n +1) , f ( n +1) (cid:105) = (cid:104) P ( n +1) ( ω ) , f ( n +1) (cid:105) − n (cid:88) k =0 (cid:104) ω ⊗ k , V k,n +1 f ( n +1) (cid:105) (cid:104) P ( n +1) ( ω ) , f ( n +1) (cid:105) − n (cid:88) k =0 (cid:32) (cid:104) P ( k ) ( ω ) , V k,n +1 f ( n +1) (cid:105) + k − (cid:88) i =0 (cid:104) P ( i ) ( ω ) , R i,k V k,n +1 f ( n +1) (cid:105) (cid:33) , which implies the statement for n + 1.(ii) This follows immediately from (i). Definition . Let ( P ( n ) ) ∞ n =0 be a monic polynomial sequence on Φ (cid:48) . For each ζ ∈ Φ (cid:48) ,we define a lowering operator Q ( ζ ) as the linear operator on P (Φ (cid:48) ) (cf. (3.5)) satisfying Q ( ζ ) (cid:104) P ( n ) , f ( n ) (cid:105) := (cid:104) P ( n − , A ( ζ ) f ( n ) (cid:105) , f ( n ) ∈ Φ (cid:12) n , n ∈ N ,Q ( ζ ) (cid:104) P (0) , f (0) (cid:105) := 0 , f (0) ∈ R , where the operator A ( ζ ) is defined by Definition 2.17. Lemma 3.7.
For every ζ ∈ Φ (cid:48) , we have Q ( ζ ) ∈ L ( P (Φ (cid:48) )) .Proof. We define an operator R ∈ L ( F fin (Φ)) by setting, for each f ( n ) ∈ Φ (cid:12) n , (cid:0) Rf ( n ) (cid:1) ( k ) := R k,n f ( n ) , k < nf ( n ) , k = n, , k > n, where the operators R k,n are as in (3.4). Similarly, using the operators V k,n fromformula (3.3), we define an operator V ∈ L ( F fin (Φ)). As easily seen, Q ( ζ ) = IV A ( ζ ) RI − , where I is the homeomorphism defined by (2.5). This implies the required result.The simplest example of a monic polynomial sequence on Φ (cid:48) is P ( n ) ( ω ) = ω ⊗ n , n ∈ N . In this case, for f ( n ) ∈ Φ (cid:12) n , (cid:104) P ( n ) ( ω ) , f ( n ) (cid:105) = (cid:104) ω ⊗ n , f ( n ) (cid:105) is just a monomial on Φ (cid:48) of degree n . For each ζ ∈ Φ (cid:48) , we obviously have Q ( ζ ) = D ( ζ ), i.e., the correspondinglowering operators are just differentiation operators. Furthermore, we trivially see inthis case that, for any n ∈ N and any ω, ζ ∈ Φ (cid:48) , P ( n ) ( ω + ζ ) = n (cid:88) k =0 (cid:18) nk (cid:19) P ( k ) ( ω ) (cid:12) P ( n − k ) ( ζ ) . (3.6) Definition . Let ( P ( n ) ) ∞ n =0 be a monic polynomial sequence on Φ (cid:48) . We say that( P ( n ) ) ∞ n =0 is of binomial type if, for any n ∈ N and any ω, ζ ∈ Φ (cid:48) , formula (3.6) holds. Remark . A monic polynomial sequence ( P ( n ) ) ∞ n =0 is of binomial type if and only if,for any n ∈ N , ω, ζ ∈ Φ (cid:48) , and ξ ∈ Φ, (cid:104) P ( n ) ( ω + ζ ) , ξ ⊗ n (cid:105) = n (cid:88) k =0 (cid:18) nk (cid:19) (cid:104) P ( k ) ( ω ) , ξ ⊗ k (cid:105)(cid:104) P ( n − k ) ( ζ ) , ξ ⊗ ( n − k ) (cid:105) . Lemma 3.10.
Let ( P ( n ) ) ∞ n =0 be a monic polynomial sequence on Φ (cid:48) of binomial type.Then, for each n ∈ N , P ( n ) (0) = 0 .Proof. We proceed by induction on n . For n = 1, it follows from (3.6) that P (1) ( ω + ζ ) = P (1) ( ω ) + P (1) ( ζ ) , ω, ζ ∈ Φ (cid:48) . Setting ζ = 0, one obtains P (1) (0) = 0. Assume that the statement holds for 1 , , . . . n .Then, for all ω, ζ ∈ Φ (cid:48) , P ( n +1) ( ω + ζ ) = P ( n +1) ( ω ) + n (cid:88) k =1 (cid:18) n + 1 k (cid:19) P ( k ) ( ω ) (cid:12) P ( n +1 − k ) ( ζ ) + P ( n +1) ( ζ ) . Setting ζ = 0, we conclude P ( n +1) (0) = 0. Our next aim is to derive equivalent characterizations of a polynomial sequence on Φ (cid:48) of binomial type. As mentioned in Introduction, it will be important for our consider-ations that Φ (cid:48) will be chosen as a space of generalized functions.So we fix d ∈ N and choose Φ to be the nuclear space D := C ∞ ( R d ) of all real-valuedsmooth functions on R d with compact support. More precisely, let T denote the setof all pairs ( l, ϕ ) with l ∈ N and ϕ ∈ C ∞ ( R d ), ϕ ( x ) ≥ x ∈ R d . For each τ = ( l, ϕ ) ∈ T , we denote by H τ the Sobolev space W l, ( R d , ϕ ( x ) dx ). Then D = proj lim τ ∈ T H τ , see [8, Chapter 14, Subsec. 4.3] for details. As the center space H = H τ we choose L ( R d , dx ) (i.e., τ = (0 , D ⊂ L ( R d , dx ) ⊂ D (cid:48) . Note that nuclear space D (cid:12) n consists of all functions from C ∞ (( R d ) n ) that aresymmetric in the variables ( x , . . . , x n ) ∈ ( R d ) n .For each x ∈ R d , the delta function δ x belongs to D (cid:48) , and we will use the notations D ( x ) := D ( δ x ) , E ( x ) := E ( δ x ) , Q ( x ) := Q ( δ x )(the latter operator being defined for a given fixed monic polynomial sequence on D (cid:48) ).Below, for any F ( k ) ∈ D (cid:48)(cid:12) k and f ( k ) ∈ D (cid:12) k , k ∈ N , we denote (cid:104) F ( k ) ( x , . . . , x k ) , f ( k ) ( x , . . . , x k ) (cid:105) := (cid:104) F ( k ) , f ( k ) (cid:105) . heorem 4.1. Let ( P ( n ) ) ∞ n =0 be a monic polynomial sequence on D (cid:48) such that P ( n ) (0) =0 for all n ∈ N . Let ( Q ( ζ )) ζ ∈D (cid:48) be the corresponding lowering operators. Then thefollowing conditions are equivalent: (BT1) The sequence ( P ( n ) ) ∞ n =0 is of binomial type. (BT2) For each ζ ∈ D (cid:48) , Q ( ζ ) is shift-invariant. (BT3) There exists a sequence ( B k ) ∞ k =1 with B k ∈ L ( D (cid:48) , D (cid:48)(cid:12) k ) , k ≥ and B = , theidentity operator on D (cid:48) , such that for all ζ ∈ D (cid:48) and P ∈ P ( D (cid:48) ) , ( Q ( ζ ) P )( ω ) = ∞ (cid:88) k =1 k ! (cid:10) ( B k ζ )( x , . . . , x k ) , ( D ( x ) · · · D ( x k ) P )( ω ) (cid:11) , ω ∈ D (cid:48) . (4.1)(BT4) The monic polynomial sequence ( P ( n ) ) ∞ n =0 has the generating function ∞ (cid:88) n =0 n ! (cid:104) P ( n ) ( ω ) , ξ ⊗ n (cid:105) = ∞ (cid:88) n =0 n ! (cid:104) ω ⊗ n , A ( ξ ) ⊗ n (cid:105) = ∞ (cid:88) n =0 n ! (cid:104) ω, A ( ξ ) (cid:105) n = exp (cid:2) (cid:104) ω, A ( ξ ) (cid:105) (cid:3) , ω ∈ D (cid:48) . (4.2) Here A ( ξ ) = ∞ (cid:88) k =1 A k ξ ⊗ k ∈ S ( D , D ) , (4.3) where A k ∈ L ( D (cid:12) k , D ) , k ≥ , and A = , the identity operator on D , while (4.2) is an equality in S ( D , R ) .Remark . Note that (cid:80) ∞ n =0 1 n ! (cid:104) ω ⊗ n , A ( ξ ) ⊗ n (cid:105) ∈ S ( D , R ) in formula (4.2) is the com-position of exp[ (cid:104) ω, ξ (cid:105) ] := (cid:80) ∞ n =0 1 n ! (cid:104) ω ⊗ n , ξ ⊗ n (cid:105) ∈ S ( D , R ) and A ( ξ ) ∈ S ( D , D ). Remark . Let A ( ξ ) ∈ S ( D , D ) be as in (4.2). Denote B ( ξ ) := ∞ (cid:88) k =1 k ! B ∗ k ξ ⊗ k ∈ S ( D , D ) , where the operators B k are as in (BT3). It will follow from the proof of Theorem 4.1that B ( ξ ) is the compositional inverse of A ( ξ ), see Definition A.9, Proposition A.11,and Remark A.12. Remark . It will also follow from the proof of Theorem 4.1 that, in (BT3), for each k ≥
2, we have B k = R ∗ ,k , the adjoint of the operator R ,k ∈ L ( D (cid:12) k , D ) from formula(3.4). 16efore proving this theorem, let us first note its immediate corollary. Corollary 4.5.
Consider any sequence ( A k ) ∞ k =1 with A k ∈ L ( D (cid:12) k , D ) , k ≥ , and A = . Then there exists a unique sequence ( P ( n ) ) ∞ n =0 of monic polynomials on D (cid:48) ofbinomial type that has the generating function (4.2) with A ( ξ ) given by (4.3) .Proof. Define A ( ξ ) ∈ S ( D , D ) by formula (4.3). For each ω ∈ D (cid:48) , define ( n ! P ( n ) ( ω )) ∞ n =0 ∈F ( D (cid:48) ) by formula (4.2). It easily follows from Definitions 3.2 and A.5 that ( P ( n ) ) ∞ n =0 is amonic polynomial sequence on D (cid:48) . Furthermore, for n ∈ N , in the representation (3.2)of P ( n ) ( ω ), we obtain U n, = 0 so that P ( n ) (0) = 0. Now the statement follows fromTheorem 4.1.We will now prove Theorem 4.1. Proof of (BT1) ⇒ (BT2) . First, we note that, for any η, ζ ∈ D (cid:48) , E ( ζ ) Q ( η )1 = Q ( η ) E ( ζ )1 = 0 . (4.4)Next, using the the binomial identity (3.6), we get, for all ξ ∈ D and n ∈ N , Q ( η ) E ( ζ ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) = n (cid:88) k =0 (cid:18) nk (cid:19) (cid:104) P ( n − k ) ( ζ ) , ξ ⊗ ( n − k ) (cid:105) Q ( η ) (cid:104) P ( k ) , ξ ⊗ k (cid:105) = (cid:104) η, ξ (cid:105) n (cid:88) k =1 (cid:18) nk (cid:19) k (cid:104) P ( n − k ) ( ζ ) , ξ ⊗ ( n − k ) (cid:105)(cid:104) P ( k − , ξ ⊗ ( k − (cid:105) = n (cid:104) η, ξ (cid:105) n (cid:88) k =1 (cid:18) n − k − (cid:19) (cid:104) P ( n − k ) ( ζ ) , ξ ⊗ ( n − k ) (cid:105)(cid:104) P ( k − , ξ ⊗ ( k − (cid:105) = n (cid:104) η, ξ (cid:105) n − (cid:88) k =0 (cid:18) n − k (cid:19) (cid:104) P ( n − k − ( ζ ) , ξ ⊗ ( n − k − (cid:105)(cid:104) P ( k ) , ξ ⊗ k (cid:105) = n (cid:104) η, ξ (cid:105) E ( ζ ) (cid:104) P ( n − , ξ ⊗ ( n − (cid:105) = E ( ζ ) Q ( η ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) . (4.5)By (4.4), (4.5), and Lemma 3.5, we get E ( ζ ) Q ( η ) P = Q ( η ) E ( ζ ) P for all P ∈ P ( D (cid:48) ).In order to prove the implication (BT2) ⇒ (BT1), we first need the following Proposition 4.6 (Polynomial expansion) . Let ( P ( n ) ) ∞ n =0 be a monic polynomial se-quence on D (cid:48) such that P ( n ) (0) = 0 for all n ∈ N , or, equivalently, for P ( n ) being of theform (3.2) , U n, = 0 . Let ( Q ( ζ )) ζ ∈D (cid:48) be the corresponding lowering operators. Then,for each P ∈ P ( D (cid:48) ) , we have P ( ω ) = ∞ (cid:88) k =0 k ! (cid:10) P ( k ) ( ω )( x , . . . , x k ) , ( Q ( x ) · · · Q ( x k ) P )(0) (cid:11) , ω ∈ D (cid:48) . (4.6)17 ere, for k = 0 , we set Q ( x ) · · · Q ( x k ) P := P .Proof. For x , . . . , x k ∈ R d , k ∈ N , ξ ∈ D , and n ∈ N , we have Q ( x ) · · · Q ( x k ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) = ( n ) k ξ ( x ) · · · ξ ( x k ) (cid:104) P ( n − k ) , ξ ⊗ ( n − k ) (cid:105) , (4.7)where ( n ) k := n ( n − · · · ( n − k + 1). Note that ( n ) k = 0 for k > n . Hence, for k, n ∈ N , one finds (cid:0) Q ( x ) · · · Q ( x k ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) (cid:1) (0) = δ k,n n ! ξ ⊗ n ( x , . . . , x n ) , where δ k,n denotes the Kronecker symbol. Thus, (cid:104) P ( n ) ( ω ) , ξ ⊗ n (cid:105) = ∞ (cid:88) k =0 k ! (cid:10) P ( k ) ( ω )( x , . . . , x k ) , (cid:0) Q ( x ) · · · Q ( x k ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) (cid:1) (0) (cid:11) . Hence, by Lemma 3.5, formula (4.6) holds for a generic P ∈ P ( D (cid:48) ). Proof of (BT2) ⇒ (BT1) . Let ζ ∈ D (cid:48) , ξ ∈ D , and n ∈ N . An application of Proposi-tion 4.6 to the polynomial P = E ( ζ ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) yields (cid:104) P ( n ) ( ω + ζ ) , ξ ⊗ n (cid:105) = ∞ (cid:88) k =0 k ! (cid:10) P ( k ) ( ω )( x , . . . , x k ) , ( Q ( x ) · · · Q ( x k ) E ( ζ ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) )(0) (cid:11) , (4.8)and by (BT2) and (4.7), we have, for k ∈ N , (cid:0) Q ( x ) · · · Q ( x k ) E ( ζ ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) (cid:1) (0) = (cid:0) E ( ζ ) Q ( x ) · · · Q ( x k ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) (cid:1) (0)= (cid:0) Q ( x ) · · · Q ( x k ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) (cid:1) ( ζ )= ( n ) k ζ ( x ) · · · ζ ( x k ) (cid:104) P ( n − k ) ( ζ ) , ξ ⊗ ( n − k ) (cid:105) . Hence, (cid:104) P ( n ) ( ω + ζ ) , ξ ⊗ n (cid:105) = n (cid:88) k =0 (cid:18) nk (cid:19) (cid:104) P ( k ) ( ω ) , ξ ⊗ k (cid:105)(cid:104) P ( n − k ) ( ζ ) , ξ ⊗ ( n − k ) (cid:105) . Thus, we have proved the equivalence of (BT1) and (BT2). To continue the proofof Theorem 4.1, we will need the following result.
Theorem 4.7 (Operator expansion theorem) . Let ( P ( n ) ) ∞ n =0 be a monic polynomialsequence on D (cid:48) of binomial type, and let ( Q ( ζ )) ζ ∈D (cid:48) be the corresponding loweringoperators. A linear operator T acting on P ( D (cid:48) ) is continuous and shift-invariant ifand only if there is a ( G ( k ) ) ∞ k =0 ∈ F ( D (cid:48) ) such that, for each P ∈ P ( D (cid:48) ) , ( T P )( ω ) = ∞ (cid:88) k =0 k ! (cid:104) G ( k ) ( x , . . . , x k ) , ( Q ( x ) · · · Q ( x k ) P )( ω ) (cid:105) , ω ∈ D (cid:48) . (4.9)18 n the latter case, for each k ∈ N and f ( k ) ∈ D (cid:12) k , (cid:104) G ( k ) , f ( k ) (cid:105) = (cid:0) T (cid:104) P ( k ) , f ( k ) (cid:105) (cid:1) (0) . (4.10) Remark . Below we will sometimes write formula (4.9) in the form T = ∞ (cid:88) k =0 k ! (cid:10) G ( k ) ( x , . . . , x k ) , Q ( x ) · · · Q ( x k ) (cid:11) . Proof of Theorem 4.7.
Let n ∈ N , ω, ζ ∈ D (cid:48) , and ξ ∈ D . By (4.8), we have (cid:0) E ( ζ ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) (cid:1) ( ω ) = ∞ (cid:88) k =0 k ! (cid:10) P ( k ) ( ω )( x , . . . , x k ) , ( Q ( x ) · · · Q ( x k ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) )( ζ ) (cid:11) . (4.11)Note that formula (4.11) remains true when n = 0. Hence, by Lemma 3.5, for each P ∈ P ( D (cid:48) ), (cid:0) E ( ζ ) P (cid:1) ( ω ) = ∞ (cid:88) k =0 k ! (cid:10) P ( k ) ( ω )( x , . . . , x k ) , ( Q ( x ) · · · Q ( x k ) P )( ζ ) (cid:11) . (4.12)Assume T ∈ L ( P ( D (cid:48) )) is shift-invariant. Swapping ω and ζ in (4.12) and applying T to this equality, we get, for any ω, ζ ∈ D (cid:48) and P ∈ P ( D (cid:48) ),( T E ( ω ) P )( ζ ) = ∞ (cid:88) k =0 k ! (cid:0) T (cid:10) P ( k ) ( · )( x , . . . , x k ) , ( Q ( x ) · · · Q ( x k ) P )( ω ) (cid:11)(cid:1) ( ζ ) . (4.13)By shift-invariance, the left hand side of (4.13) is equal to ( T P )( ω + ζ ). In particular,this holds for ζ = 0:( T P )( ω ) = ∞ (cid:88) k =0 k ! (cid:0) T (cid:10) P ( k ) ( · )( x , . . . , x k ) , ( Q ( x ) · · · Q ( x k ) P )( ω ) (cid:11)(cid:1) (0) . (4.14)Let G ( k ) ∈ D (cid:48)(cid:12) k , k ∈ N , be defined by (4.10). Then (4.9) follows from (4.14).Conversely, let ( G ( k ) ) ∞ k =0 ∈ F ( D (cid:48) ) be fixed, and let T be given by (4.9). As easilyseen, T ∈ L ( P ( D (cid:48) )). For each P ∈ P ( D (cid:48) ) and ζ ∈ D (cid:48) , we get from (4.9) and (BT2):( T E ( ζ ) P )( ω ) = ∞ (cid:88) k =0 k ! (cid:10) G ( k ) ( x , . . . , x k ) , ( E ( ζ ) Q ( x ) · · · Q ( x k ) P )( ω ) (cid:11) = ∞ (cid:88) k =0 k ! (cid:10) G ( k ) ( x , . . . , x k ) , ( Q ( x ) · · · Q ( x k ) P )( ω + ζ ) (cid:11) = ( T P )( ω + ζ ) = ( E ( ζ ) T P )( ω ) . Therefore, the operator T is shift-invariant. Moreover, it easily follows from (4.9) and(4.7) that (4.10) holds. 19ote that the statement (BT3) ⇒ (BT2) follows immediately from Theorem 4.7. Proof of (BT2) ⇒ (BT3) . Let ζ ∈ D (cid:48) . We apply Theorem 4.7 to the sequence ofmonomials and its family of lowering operators, ( D ( η )) η ∈D (cid:48) , and the shift-invariantoperator Q ( ζ ). By using also formula (3.4), we obtain Q ( ζ ) = ∞ (cid:88) k =1 k ! (cid:10) G ( k ) ( ζ, x , . . . , x k ) , D ( x ) · · · D ( x k ) (cid:11) , (4.15)where (cid:104) G ( k ) ( ζ, x , . . . , x k ) , f ( k ) ( x , . . . , x k ) (cid:105) = (cid:0) Q ( ζ ) (cid:104)· ⊗ k , f ( k ) (cid:105) (cid:1) (0) = (cid:104) ζ, R ,k f ( k ) (cid:105) (4.16)for all k ∈ N and f ( k ) ∈ D (cid:12) k . Here, we set R , := , the identity operator on D . For k ∈ N , we denote B k := R ∗ ,k ∈ L ( D (cid:48) , D (cid:48)(cid:12) k ). Note that B = , the identity operatoron D (cid:48) . By (4.16), G ( k ) ( ζ, · ) = B k ζ, k ≥ . (4.17)Formulas (4.15), (4.17) imply (BT3).Thus, we have proved the equivalence of (BT1), (BT2), and (BT3).According to Theorem 4.7, under the conditions assumed therein, there is a one-to-one correspondence between shift-invariant operators T and sequences ( k ! G ( k ) ) ∞ k =0 ∈F ( D (cid:48) ). We noted above that the space S ( P ( D (cid:48) )) of shift-invariant operators is analgebra under the product of operators, while F ( D (cid:48) ) is a commutative algebra underthe symmetric tensor product. Theorem 4.9 (The isomorphism theorem) . Let ( P ( n ) ) ∞ n =0 be a sequence of monic poly-nomials on D (cid:48) of binomial type, and let ( Q ( ζ )) ζ ∈D (cid:48) be the corresponding lowering op-erators. Then, the correspondence given by Theorem 4.7, S ( P ( D (cid:48) )) (cid:51) T (cid:55)→ J T := (cid:18) k ! G ( k ) (cid:19) ∞ k =0 ∈ F ( D (cid:48) ) , is an algebra isomorphism.Proof. In view of Theorem 4.7, we only have to prove that, for any
S, T ∈ S ( P ( D (cid:48) )), J ( ST ) = J S (cid:12)
J T. (4.18)Let
J S = (cid:18) k ! F ( k ) (cid:19) ∞ k =0 , J T = (cid:18) k ! G ( k ) (cid:19) ∞ k =0 .
20y Theorem 4.7, for all ξ ∈ D and ω ∈ D (cid:48) , (cid:0) T (cid:104) P ( n ) , ξ ⊗ n (cid:105) (cid:1) ( ω ) = n (cid:88) k =0 (cid:18) nk (cid:19) (cid:104) G ( k ) , ξ ⊗ k (cid:105)(cid:104) P ( n − k ) ( ω ) , ξ ⊗ ( n − k ) (cid:105) , (4.19)and a similar expression holds for S . Therefore, (cid:0) ST (cid:104) P ( n ) , ξ ⊗ n (cid:105) (cid:1) ( ω )= n (cid:88) k =0 (cid:18) nk (cid:19) (cid:104) G ( k ) , ξ ⊗ k (cid:105) (cid:0) S (cid:104) P ( n − k ) , ξ ⊗ ( n − k ) (cid:105) (cid:1) ( ω )= n (cid:88) k =0 (cid:18) nk (cid:19) (cid:104) G ( k ) , ξ ⊗ k (cid:105) n − k (cid:88) i =0 (cid:18) n − ki (cid:19) (cid:104) F ( i ) , ξ ⊗ i (cid:105)(cid:104) P ( n − k − i ) ( ω ) , ξ ⊗ ( n − k − i ) (cid:105) = n (cid:88) k =0 n − k (cid:88) i =0 n ! k ! i ! ( n − k − i )! (cid:104) G ( k ) (cid:12) F ( i ) , ξ ⊗ ( k + i ) (cid:105)(cid:104) P ( n − k − i ) ( ω ) , ξ ⊗ ( n − k − i ) (cid:105) = n (cid:88) j =0 (cid:18) nj (cid:19) (cid:42) j (cid:88) k =0 (cid:18) jk (cid:19) G ( k ) (cid:12) F ( j − k ) , ξ ⊗ j (cid:43) (cid:104) P ( n − j ) ( ω ) , ξ ⊗ ( n − j ) (cid:105) . From here and (4.19), formula (4.18) follows.As an immediate consequence of Theorem 4.9, we conclude
Corollary 4.10.
Any two shift-invariant operators commute.
Corollary 4.11.
Let the conditions of Theorem 4.9 be satisfied and let the operator J : S ( P ( D (cid:48) )) → F ( D (cid:48) ) be defined as in that theorem. Define J : S ( P ( D (cid:48) )) → S ( D , R ) by J := I J . Here I : F ( D (cid:48) ) → S ( D , R ) is defined by (2.11) . Then, for each T ∈ S ( P ( D (cid:48) )) , we have ( J T )( ξ ) = ∞ (cid:88) n =0 n ! (cid:0) T (cid:104) P ( n ) , ξ ⊗ n (cid:105) (cid:1) (0) . (4.20) Furthermore, J is an algebra isomorphism, i.e., for any S, T ∈ S ( P ( D (cid:48) )) , we have (cid:0) J ( ST ) (cid:1) ( ξ ) = ( J S )( ξ )( J T )( ξ ) . (4.21) Proof.
Formula (4.20) follows Theorem 4.7 and the definition of J . Formula (4.21) isa consequence of (2.12) and Theorem 4.9. Corollary 4.12.
Let T ∈ S ( P ( D (cid:48) )) . The operator T is invertible if and only if T (cid:54) = 0 .Furthermore, if T (cid:54) = 0 , then T − ∈ S ( P ( D (cid:48) )) . roof. If T T is not equal to { } . Hence, T is not invertible.Assume T (cid:54) = 0. Let the isomorphism J from Theorem 4.9 be constructed throughthe monomials and the corresponding lowering operators D ( ζ ), ζ ∈ D (cid:48) . So formula(4.20) becomes ( J T )( ξ ) = ∞ (cid:88) n =0 n ! (cid:0) T (cid:104)· ⊗ n , ξ ⊗ n (cid:105) (cid:1) (0) . (4.22)Since T (cid:54) = 0, formula (4.22), Corollary 4.11, and Proposition A.1 imply the existenceof an operator S ∈ S ( P ( D (cid:48) )) such that ST = T S = . Hence, the operator T isinvertible and T − = S ∈ S ( P ( D (cid:48) )). Proof of (BT3) ⇒ (BT4) . Let the isomorphism J from Theorem 4.9 be constructedthrough the monomials and the corresponding lowering operators D ( ζ ), ζ ∈ D (cid:48) . By(4.22), ( J D ( ζ ))( ξ ) = (cid:104) ζ, ξ (cid:105) . Thus, by Lemma 2.20 and the isomorphism theorem,( J E ( ζ ))( ξ ) = ∞ (cid:88) n =0 n ! ( J D ( ζ ) n )( ξ ) = ∞ (cid:88) n =0 n ! (cid:104) ζ ⊗ n , ξ ⊗ n (cid:105) = exp[ (cid:104) ζ, ξ (cid:105) ] . (4.23)Let G ( k ) ∈ D (cid:48)(cid:12) k , k ∈ N . Then formula (4.22) with T = (cid:10) G ( k ) ( x , . . . , x k ) , D ( x ) · · · D ( x k ) (cid:11) yields ( J (cid:10) G ( k ) ( x , . . . , x k ) , D ( x ) · · · D ( x k ) (cid:11) )( ξ ) = (cid:104) G ( k ) , ξ ⊗ k (cid:105) . Therefore, condition (BT3) gives, for each ζ ∈ D (cid:48) ,( J Q ( ζ ))( ξ ) = ∞ (cid:88) k =1 k ! (cid:104) B k ζ, ξ ⊗ k (cid:105) = ∞ (cid:88) k =1 k ! (cid:104) ζ, R ,k ξ ⊗ k (cid:105) . (4.24)In the latter equality we used the fact that R ∗ ,k = B k , see the proof of (BT2) ⇒ (BT3).Choosing in (4.24) ζ = δ x , x ∈ R d , we obtain( J Q ( x ))( ξ ) = ∞ (cid:88) k =1 k ! (cid:0) R ,k ξ ⊗ k (cid:1) ( x ) , and, more generally, by Theorem 4.9, for any x , . . . , x k ∈ R d , k ∈ N ,( J Q ( x ) · · · Q ( x k ))( ξ ) = k (cid:89) i =1 (cid:32) ∞ (cid:88) n =1 n ! (cid:0) R ,n ξ ⊗ n (cid:1) ( x i ) (cid:33) . (4.25)22y (4.22) and (4.25), we get, for each G ( k ) ∈ D (cid:48)(cid:12) k , k ∈ N , (cid:0) J (cid:104) G ( k ) ( x , . . . , x k ) , Q ( x ) · · · Q ( x k ) (cid:105) (cid:1) ( ξ )= (cid:10) G ( k ) ( x , . . . , x k ) , J ( Q ( x ) · · · Q ( x k ))( ξ ) (cid:11) = (cid:42) G ( k ) ( x , . . . , x k ) , k (cid:89) i =1 (cid:32) ∞ (cid:88) n =1 n ! (cid:0) R ,n ξ ⊗ n (cid:1) ( x i ) (cid:33)(cid:43) = (cid:42) G ( k ) , (cid:32) ∞ (cid:88) n =1 n ! R ,n ξ ⊗ n (cid:33) ⊗ k (cid:43) . (4.26)By (4.12) and (4.26),( J E ( ζ ))( ξ ) = ∞ (cid:88) k =0 k ! (cid:42) P ( k ) ( ζ ) , (cid:32) ∞ (cid:88) n =1 n ! R ,n ξ ⊗ n (cid:33) ⊗ k (cid:43) . (4.27)Formulas (4.23) and (4.27) imply ∞ (cid:88) k =0 k ! (cid:42) P ( k ) ( ζ ) , (cid:32) ∞ (cid:88) n =1 n ! R ,n ξ ⊗ n (cid:33) ⊗ k (cid:43) = exp[ (cid:104) ζ, ξ (cid:105) ] . (4.28)By Proposition A.11, we find the compositional inverse A ( ξ ) = (cid:80) ∞ k =1 A k ξ ⊗ k ∈S ( D , D ) of (cid:80) ∞ n =1 1 n ! R ,n ξ ⊗ n ∈ S ( D , D ), and by Remark A.12, we have A = . For-mula (4.2) now follows from (4.28) and Proposition A.7. Remark . It follows from the proof of Proposition A.11 that, for k ≥
2, the opera-tors A k are given by the recurrence formula A k = − k (cid:88) n =2 n ! R ,n (cid:88) ( l ,...,l n ) ∈ N n l + ··· + l n = k A l (cid:12) · · · (cid:12) A l n . (4.29) Proof of (BT4) ⇒ (BT1) . By (BT4), we have, for any ω, ζ ∈ D (cid:48) , ∞ (cid:88) n =0 n ! (cid:104) P ( n ) ( ω + ζ ) , ξ ⊗ n (cid:105) = (cid:32) ∞ (cid:88) n =0 n ! (cid:104) P ( n ) ( ω ) , ξ ⊗ n (cid:105) (cid:33) (cid:32) ∞ (cid:88) k =0 k ! (cid:104) P ( k ) ( ζ ) , ξ ⊗ k (cid:105) (cid:33) = ∞ (cid:88) n =0 n ! (cid:42) n (cid:88) k =0 (cid:18) nk (cid:19) P ( k ) ( ω ) (cid:12) P ( n − k ) ( ζ ) , ξ ⊗ n (cid:43) , which implies (BT1). This concludes the proof of Theorem 4.1.23 efinition . Let ( Q ( ζ )) ζ ∈D (cid:48) be a family of operators from L ( P ( D (cid:48) )). We say that( Q ( ζ )) ζ ∈D (cid:48) is a family of delta operators if the following conditions are satisfied:(i) Each Q ( ζ ) is shift-invariant;(ii) For each ζ ∈ D (cid:48) and each ξ ∈ D , Q ( ζ ) (cid:104)· , ξ (cid:105) = (cid:104) ζ, ξ (cid:105) ; (4.30)(iii) Q ( ζ ) linearly depends on ζ ∈ D (cid:48) . Furthermore, for each k ≥
2, the mapping D (cid:48) (cid:51) ζ (cid:55)→ B k ζ ∈ D (cid:48)(cid:12) k defined by (cid:104) B k ζ, f ( k ) (cid:105) := (cid:0) Q ( ζ ) (cid:104)· ⊗ k , f ( k ) (cid:105) (cid:1) (0) , f ( k ) ∈ D (cid:12) k , (4.31)belongs to L ( D (cid:48) , D (cid:48)(cid:12) k ).It is a straightforward consequence of Theorem 4.1 that, for any monic polynomialsequence ( P ( n ) ) ∞ n =0 of binomial type, the corresponding family ( Q ( ζ )) ζ ∈D (cid:48) of loweringoperators is a family of delta operators. Proposition 4.15.
Let ( Q ( ζ )) ζ ∈D (cid:48) be a family of delta operators. Then, there exists aunique monic polynomial sequence ( P ( n ) ) ∞ n =0 of binomial type for which ( Q ( ζ )) ζ ∈D (cid:48) isthe family of lowering operators.Proof. Let ζ, ω ∈ D (cid:48) and ξ ∈ D . By shift-invariance of Q ( ζ ) and (4.30), we get (cid:104) ζ, ξ (cid:105) = E ( ω ) (cid:104) ζ, ξ (cid:105) = E ( ω ) Q ( ζ ) (cid:104)· , ξ (cid:105) = Q ( ζ ) E ( ω ) (cid:104)· , ξ (cid:105) = Q ( ζ ) (cid:104)· + ω, ξ (cid:105) = Q ( ζ ) (cid:104)· , ξ (cid:105) + (cid:104) ω, ξ (cid:105) Q ( ζ )1 = (cid:104) ζ, ξ (cid:105) + (cid:104) ω, ξ (cid:105) Q ( ζ )1 , which implies that Q ( ζ )1 = 0. Hence, by Theorem 4.7, (4.30), and (4.31), we have Q ( ζ ) = ∞ (cid:88) k =1 k ! (cid:10) ( B k ζ )( x , . . . , x k ) , D ( x ) · · · D ( x k ) (cid:105) , with B := , the identity operator on D (cid:48) .Let A := be the identity operator on D , and for k ≥
2, let the operators A k ∈L ( D (cid:12) k , D ) be defined by the recurrence formula (4.29) with R ,k := B ∗ k . Thus, A ( ξ ) = (cid:80) ∞ k =1 A k ξ ⊗ k ∈ S ( D , D ) is the compositional inverse of (cid:80) ∞ n =1 1 n ! B ∗ n ξ ⊗ n ∈ S ( D , D )Let ( P ( n ) ) ∞ n =0 be the monic polynomial sequence on D (cid:48) of binomial type that has thegenerating function (4.2) with A ( ξ ) given by (4.3), see Corollary 4.5. By Remark 4.3,( P ( n ) ) ∞ n =0 is the unique required polynomial sequence. Definition . Let ( Q ( ζ )) ζ ∈D (cid:48) be a family of delta operators. The correspondingmonic polynomial sequence ( P ( n ) ) ∞ n =0 of binomial type given by Proposition 4.15 iscalled the basic sequence for ( Q ( ζ )) ζ ∈D (cid:48) . Proposition 4.17. ( Q ( ζ )) ζ ∈D (cid:48) is a family of delta operators if and only if there existsa sequence ( B k ) ∞ k =1 , with B k ∈ L ( D (cid:48) , D (cid:48)(cid:12) k ) , such that B = and (4.1) holds.Proof. The statement follows immediately from Theorem 4.1 and Proposition 4.15.24
Lifting of polynomials on R of binomial type Let ( p n ) ∞ n =0 be a monic polynomial sequence on R of binomial type, and let Q be itsdelta operator, that is, Qp n = np n − for each n ∈ N . According to the one-dimensional(classical) version of Theorem 4.1 (see e.g. [24]), Q has a formal expansion Q = ∞ (cid:88) k =1 b k k ! D k = q ( D ) , (5.1)where ( b k ) k ∈ N is a sequence of real numbers such that b = 1, D is the differentiationoperator and q ( t ) := ∞ (cid:88) k =1 b k k ! t k (5.2)is a formal power series in t ∈ R . Furthermore, ∞ (cid:88) n =0 u n n ! p n ( t ) = exp[ ta ( u )] , (5.3)where the formal power series in u ∈ R , a ( u ) = ∞ (cid:88) k =1 a k u k (5.4)is the compositional inverse of q . In particular, a = 1. We will now lift the sequenceof polynomials ( p n ) ∞ n =0 to a monic polynomial sequence on D (cid:48) of binomial type.For each k ∈ N , we define an operator D k ∈ L ( D (cid:12) k , D ) by( D k f ( k ) )( x ) := f ( k ) ( x, . . . , x ) , f ( k ) ∈ D (cid:12) k , x ∈ R d (5.5)( D being the identity operator on D ). The adjoint operator D ∗ k ∈ L ( D (cid:48) , D (cid:48)(cid:12) k ) satisfies (cid:104) D ∗ k ζ, f ( k ) (cid:105) = (cid:104) ζ ( x ) , f ( k ) ( x, . . . , x ) (cid:105) , ζ ∈ D (cid:48) , f ( k ) ∈ D (cid:12) k . In particular, (cid:104) D ∗ k ζ, ξ ⊗ k (cid:105) = (cid:104) ζ, ξ k (cid:105) , ζ ∈ D (cid:48) , ξ ∈ D . (5.6)We now define an operator B k : D (cid:48) → D (cid:48)(cid:12) k by B k := b k D ∗ k , k ∈ N , (5.7)where the numbers b k are as in (5.1). Let ( Q ( ζ )) ζ ∈D (cid:48) be the family of delta operatorsgiven by (4.1), see Proposition 4.17. By (5.6) and (5.7), we then have Q ( x ) = ∞ (cid:88) k =1 b k k ! D k ( x ) = q ( D ( x )) , x ∈ R d , (5.8)25nd moreover, Q ( ζ ) = (cid:28) ζ ( x ) , ∞ (cid:88) k =1 b k k ! D k ( x ) (cid:29) = (cid:10) ζ ( x ) , q ( D ( x )) (cid:11) . (5.9)Let ( P ( n ) ) ∞ n =0 be the basic sequence for ( Q ( ζ )) ζ ∈D (cid:48) . Thus, in view of (5.1) and (5.8),we may think of ( P ( n ) ) ∞ n =0 as the lifting of the monic polynomial sequence ( p n ) ∞ n =0 ofbinomial type .As easily seen, the generating function of ( P ( n ) ) ∞ n =0 is given by (4.2) with the oper-ators A k ∈ L ( D (cid:12) k , D ) given by A k = a k D k , k ∈ N , where the numbers a k are as in (5.4). Therefore, by (4.3), A ( ξ ) = ∞ (cid:88) k =1 a k ξ k = a ( ξ ) . (5.10)Hence, ∞ (cid:88) n =0 n ! (cid:104) P ( n ) ( ω ) , ξ ⊗ n (cid:105) = exp (cid:34)(cid:42) ω ( x ) , ∞ (cid:88) k =1 a k ξ k ( x ) (cid:43)(cid:35) = exp (cid:2) (cid:104) ω, a ( ξ ) (cid:105) (cid:3) . (5.11)Thus, this generating function can be though of as the lifting of the generating func-tion (5.3).Recall that a set partition π of a set X (cid:54) = ∅ is an (unordered) collection of disjointnonempty subsets of X whose union equals X . We denote by P ( n ) the collection ofall set partitions of X = { , , . . . , n } . For a set B ⊂ { , . . . , n } , we denote by | B | thecardinality of B . Proposition 5.1.
Let ( P ( n ) ) ∞ n =0 be the monic polynomial sequence on D (cid:48) of binomialtype that has the generating function (5.11) . For k ∈ N , denote α k := a k k ! , so that a ( u ) = ∞ (cid:88) k =1 α k k ! u k . (5.12) Then, for any n ∈ N , ω ∈ D (cid:48) , and ξ ∈ D , (cid:104) P ( n ) ( ω ) , ξ ⊗ n (cid:105) = (cid:88) π ∈ P ( n ) (cid:89) B ∈ π α | B | (cid:104) ω, ξ | B | (cid:105) , (5.13) or equivalently P ( n ) ( ω ) = n (cid:88) k =1 (cid:88) { B ,B ,...,B k }∈ P ( n ) α | B | α | B | · · · α | B k | × (cid:0) D ∗| B | ω ⊗| B | (cid:1) (cid:12) (cid:0) D ∗| B | ω ⊗| B | (cid:1) (cid:12) · · · (cid:12) (cid:0) D ∗| B k | ω ⊗| B k | (cid:1) . (5.14)26 roof. It follows immediately from the form of the generating function (5.11) that (cid:104) P ( n ) ( ω ) , ξ ⊗ n (cid:105) = n (cid:88) k =1 n ! k ! (cid:88) ( i ,...,i k ) ∈ N k i + ··· + i k = n a i · · · a i k (cid:104) ω, ξ i (cid:105) · · · (cid:104) ω, ξ i k (cid:105) = (cid:88) ( j ,j ...,j n ) ∈ N n j +2 j + ··· + nj n = n n ! j ! j ! · · · j n ! (1!) j (2!) j · · · ( n !) j n α j α j · · · α j n n × (cid:104) ω, ξ (cid:105) j (cid:104) ω, ξ (cid:105) j · · · (cid:104) ω, ξ n (cid:105) j n . which implies (5.13), hence also (5.14).We denote by M ( R d ) the space of all signed Radon measures on R d , i.e., the set ofall signed measures η on ( R d , B ( R d )) such that | η | (Λ) < ∞ for all Λ ∈ B ( R d ). Here B ( R d ) denotes the Borel σ -algebra on R d , B ( R d ) denotes the collection of all boundedsets Λ ∈ B ( R d ), and | η | denotes the variation of η ∈ M ( R d ).Further, let B sym (( R d ) n ) denote the sub- σ -algebra of B (( R d ) n ) that consists of allsymmetric sets ∆ ∈ B (( R d ) n ), i.e., for each permutation σ ∈ S ( n ), ∆ is an invariantset for the mapping( R d ) n (cid:51) ( x , . . . , x n ) (cid:55)→ ( x σ (1) , . . . , x σ ( n ) ) ∈ ( R d ) n . We denote by M sym (( R d ) n ) the space of all signed Radon measures on (( R d ) n , B sym (( R d ) n )). Corollary 5.2.
Let ( P ( n ) ) ∞ n =0 be the monic polynomial sequence on D (cid:48) of binomial typethat has the generating function (5.11) . Then, for each η ∈ M ( R d ) and n ∈ N , we have P ( n ) ( η ) ∈ M sym (( R d ) n ) . Furthermore, for each Λ ∈ B ( R d ) , (cid:0) P ( n ) ( η ) (cid:1) (Λ n ) = p n ( η (Λ)) , n ∈ N , (5.15) where ( p n ) ∞ n =0 is the polynomial sequence on R with generating function (5.3) .Proof. Let η ∈ M ( R d ). For each j ∈ N , D ∗ j η ∈ M sym (( R d ) j ), since for each f ( j ) ∈ D (cid:12) j (cid:104) D ∗ j η, f ( j ) (cid:105) = (cid:90) R d f ( j ) ( x, . . . , x ) dη ( x ) . Note that the measure D ∗ j η is concentrated on the set { ( x , x , . . . , x j ) ∈ ( R d ) j | x = x = · · · = x j } . By formula (5.14), we therefore get P ( n ) ( η ) ∈ M sym (( R d ) n ).27ix any Λ ∈ B ( R d ). Set ξ := uχ Λ , where u ∈ R and χ Λ denotes the indicatorfunction of Λ. It easily follows from (5.11) by an approximation argument that ∞ (cid:88) n =0 u n n ! (cid:0) P ( n ) ( η ) (cid:1) (Λ n ) = ∞ (cid:88) n =0 n ! (cid:104) P ( n ) ( η ) , ξ ⊗ n (cid:105) = exp (cid:20)(cid:90) R d a ( ξ ( x )) dη ( x ) (cid:21) = exp [ η (Λ) a ( u )] . (5.16)In formula (5.16), (cid:104) P ( n ) ( η ) , ξ ⊗ n (cid:105) denotes the integral of the function ξ ⊗ n with respectto the measure P ( n ) ( η ). Formula (5.15) now follows from (5.3) and (5.16).The following proposition shows that the lifted polynomials ( P ( n ) ) ∞ n =0 have an ad-ditional property of binomial type. Proposition 5.3.
Let ( P ( n ) ) ∞ n =0 be the monic polynomial sequence on D (cid:48) of binomialtype that has the generating function (5.11) . Let ξ, φ ∈ D be such that { x ∈ R d | ξ ( x ) (cid:54) = 0 } ∩ { x ∈ R d | φ ( x ) (cid:54) = 0 } = ∅ . (5.17) Then, for any ω ∈ D (cid:48) and k, n ∈ N , (cid:104) P ( k + n ) ( ω ) , ξ ⊗ k (cid:12) φ ⊗ n (cid:105) = (cid:104) P ( k ) ( ω ) , ξ ⊗ k (cid:105)(cid:104) P ( n ) ( ω ) , φ ⊗ n (cid:105) . (5.18) Therefore, for each n ∈ N , (cid:104) P ( n ) ( ω ) , ( ξ + φ ) ⊗ n (cid:105) = n (cid:88) k =0 (cid:18) nk (cid:19) (cid:104) P ( k ) ( ω ) , ξ ⊗ k (cid:105)(cid:104) P ( n − k ) ( ω ) , φ ⊗ ( n − k ) (cid:105) . (5.19) Proof.
We have ∞ (cid:88) n =0 n ! (cid:104) P ( n ) ( ω ) , ( ξ + φ ) ⊗ n (cid:105) = ∞ (cid:88) n =0 n (cid:88) k =0 k !( n − k )! (cid:104) P ( n ) ( ω ) , ξ ⊗ k (cid:12) φ ⊗ ( n − k ) (cid:105) = ∞ (cid:88) k =0 ∞ (cid:88) n =0 k ! n ! (cid:104) P ( n + k ) ( ω ) , φ ⊗ n (cid:12) ξ ⊗ k (cid:105) , (5.20)and by (5.11) and (5.17) ∞ (cid:88) n =0 n ! (cid:104) P ( n ) ( ω ) , ( ξ + φ ) ⊗ n (cid:105) = exp (cid:2) (cid:104) ω, a ( ξ + φ ) (cid:105) (cid:3) = exp (cid:2) (cid:104) ω, a ( ξ ) (cid:105) (cid:3) exp (cid:2) (cid:104) ω, a ( φ ) (cid:105) (cid:3) = ∞ (cid:88) k =0 ∞ (cid:88) n =0 k ! n ! (cid:104) P ( k ) ( ω ) , ξ ⊗ k (cid:105)(cid:104) P ( n ) ( ω ) , φ ⊗ n (cid:105) . (5.21)Formulas (5.20), (5.21) imply (5.18), hence also (5.19).We will now consider examples of sequences of lifted polynomials of binomial type.28 .1 Falling factorials on D (cid:48) The classical falling factorials is the sequence ( p n ) ∞ n =0 of monic polynomials on R ofbinomial type that are explicitly given by p n ( t ) = ( t ) n := t ( t − t − · · · ( t − n + 1) . The corresponding delta operator is Q = e D −
1, so that Q is the difference operator( Qp )( t ) = p ( t + 1) − p ( t ). Here p belongs to P ( R ), the space of polynomials on R . Thegenerating function of the falling factorials is ∞ (cid:88) n =0 u n n ! ( t ) n = exp[ t log(1 + u )] = (1 + u ) t . One also defines an extension of the binomial coefficient, (cid:18) tn (cid:19) := 1 n ! ( t ) n = t ( t − t − · · · ( t − n + 1) n ! , (5.22)which becomes the classical binomial coefficient for t ∈ N , t ≥ n .Let us now consider the corresponding lifted sequence of polynomials, ( P ( n ) ) ∞ n =0 .We will call these polynomials the falling factorials on D (cid:48) . By analogy with the one-dimensional case, we will write ( ω ) n := P ( n ) ( ω ) for ω ∈ D (cid:48) .By (5.8), Q ( x ) = e D ( x ) −
1. Hence, by Boole’s formula,( Q ( x ) P )( ω ) = P ( ω + δ x ) − P ( ω ) , x ∈ R d , P ∈ P ( D (cid:48) ) , (5.23)and by (5.9),( Q ( ζ ) P )( ω ) = (cid:10) ζ ( x ) , P ( ω + δ x ) − P ( ω ) (cid:11) , ζ ∈ D (cid:48) , P ∈ P ( D (cid:48) ) . Further, by (5.11), the generating function is given by ∞ (cid:88) n =0 n ! (cid:104) ( ω ) n , ξ ⊗ n (cid:105) = exp (cid:2) (cid:104) ω, log(1 + ξ ) (cid:105) (cid:3) . (5.24) Proposition 5.4.
The falling factorials on D (cid:48) have the following explicit form: ( ω ) = 1;( ω ) = ω ;( ω ) n ( x , . . . , x n ) = ω ( x )( ω ( x ) − δ x ( x )) × ( ω ( x ) − δ x ( x ) − δ x ( x )) · · · ( ω ( x n ) − δ x ( x n ) − · · · − δ x n − ( x n )) (5.25) for n ≥ . roof. Let (( ω ) n ) ∞ n =0 denote the monic polynomial sequence on D (cid:48) defined by formula(5.25). Note that, for n ∈ N , ( ω ) n = 0 for ω = 0. It can be easily shown by inductionthat the polynomials ( ω ) n satisfy the following recurrence relation:( ω ) = 1; (cid:104) ( ω ) n +1 , ξ ⊗ ( n +1) (cid:105) = (cid:104) ( ω ) n (cid:12) ω, ξ ⊗ ( n +1) (cid:105) − n (cid:104) ( ω ) n , ξ (cid:12) ξ ⊗ ( n − (cid:105) ,ξ ∈ D , n ∈ N . (5.26)Furthermore, this recurrence relation uniquely determines the polynomials (( ω ) n ) ∞ n =0 .It suffices to prove that, for each ω ∈ D (cid:48) , n ∈ N , ξ ∈ D , and x ∈ R d , (cid:0) Q ( x ) (cid:104) ( · ) n , ξ ⊗ n (cid:105) (cid:1) ( ω ) = nξ ( x ) (cid:104) ( ω ) n − , ξ ⊗ ( n − (cid:105) , or equivalently, by (5.23), (cid:104) ( ω + δ x ) n , ξ ⊗ n (cid:105) = (cid:104) ( ω ) n , ξ ⊗ n (cid:105) + nξ ( x ) (cid:104) ( ω ) n − , ξ ⊗ ( n − (cid:105) . (5.27)We prove this formula by induction. It trivially holds for n = 1. Assume that (5.27)holds for 1 , , . . . , n and let us prove it for n + 1. Using our assumption, the recurrencerelation (5.26) and the polarization identity, we get (cid:104) ( ω + δ x ) n +1 , ξ ⊗ ( n +1) (cid:105) = (cid:104) ( ω + δ x ) n , ξ ⊗ n (cid:105)(cid:104) ω + δ x , ξ (cid:105) − n (cid:104) ( ω + δ x ) n , ξ (cid:12) ξ ⊗ ( n − (cid:105) = (cid:0) (cid:104) ( ω ) n , ξ ⊗ n (cid:105) + nξ ( x ) (cid:104) ( ω ) n − , ξ ⊗ ( n − (cid:105) (cid:1)(cid:0) (cid:104) ω, ξ (cid:105) + ξ ( x ) (cid:1) − n (cid:0) (cid:104) ( ω ) n , ξ (cid:12) ξ ⊗ ( n − (cid:105) + ξ ( x ) (cid:104) ( ω ) n − , ξ ⊗ ( n − (cid:105) + ( n − ξ ( x ) (cid:104) ( ω ) n − , ξ (cid:12) ξ ⊗ ( n − (cid:105) (cid:1) = (cid:0) (cid:104) ( ω ) n (cid:12) ω, ξ ⊗ ( n +1) (cid:105) − n (cid:104) ( ω ) n , ξ (cid:12) ξ ⊗ ( n − (cid:105) (cid:1) + ξ ( x ) (cid:104) ( ω ) n , ξ ⊗ n (cid:105) + nξ ( x ) (cid:0) (cid:104) ( ω ) n − (cid:12) ω, ξ ⊗ n (cid:105) − ( n − (cid:104) ( ω ) n − , ξ (cid:12) ξ ⊗ ( n − (cid:105) (cid:1) = (cid:104) ( ω ) n +1 , ξ ⊗ ( n +1) (cid:105) + nξ ( x ) (cid:104) ( ω ) n , ξ ⊗ n (cid:105) . Remark . Note that the recurrence relation (5.26) satisfied by the polynomials on D (cid:48) with generating function (5.24) was already discussed in [7].Since we have interpreted ( ω ) n as a falling factorial on D (cid:48) , we naturally define ‘ ω choose n ’ by (cid:0) ωn (cid:1) := n ! ( ω ) n , compare with (5.22).We denote by Γ the configuration space over R d , i.e., the space of all Radon measures γ ∈ M ( R d ) that are of the form γ = (cid:80) ∞ i =1 δ x i , where x i (cid:54) = x j if i (cid:54) = j . (We canobviously identify the configuration γ = (cid:80) ∞ i =1 δ x i with the (locally finite) set { x i } i ∈ N .)The following result is immediate. Corollary 5.6.
For each γ = (cid:80) ∞ i =1 δ x i , formula (1.1) holds. emark . Polynomials (cid:0) γn (cid:1) play a crucial role in the theory of point process (i.e.,Γ-valued random variables), see e.g. [18]. More precisely, given a probability space(Ξ , F , P ) and a point process γ : Ξ → Γ, the n th correlation measure of γ is defined asthe (unique) measure σ ( n ) on (cid:0) ( R d ) n , B sym (( R d ) n ) (cid:1) that satisfies E (cid:28)(cid:18) γn (cid:19) , f ( n ) (cid:29) = (cid:90) ( R d ) n f ( n ) ( x , . . . , x n ) dσ ( n ) ( x , . . . , x n ) for all f ( n ) ∈ D (cid:12) n , f ( n ) ≥ . Here E denotes the expectation with respect to the probability measure P . Under verymild conditions on the point process γ , the correlation measures ( σ ( n ) ) ∞ n =1 uniquelyidentify the distribution of γ on Γ. In the case where each measure σ ( n ) is absolutelycontinuous with respect to the Lebesgue measure, one defines the n th correlation func-tion of the point process γ , denote by k ( n ) ( x , . . . , x n ), as follows: dσ ( n ) ( x , . . . , x n ) = 1 n ! k ( n ) ( x , . . . , x n ) dx · · · dx n , or equivalently E (cid:10) ( γ ) n , f ( n ) (cid:11) = (cid:90) ( R d ) n f ( n ) ( x , . . . , x n ) k ( n ) ( x , . . . , x n ) dx · · · dx n . Corollary 5.8.
For each Λ ∈ B ( R d ) , γ ∈ Γ and n ∈ N , we have, (cid:18) γn (cid:19) (Λ n ) = (cid:18) γ (Λ) n (cid:19) . Proof.
The result is obvious by Corollary 5.6, or alternatively, by Corollary 5.2.
Remark . In view of Corollaries 5.6 and 5.8, for the falling factorials on D (cid:48) , the setΓ plays a role similar to that played by the set N for the falling factorials on R . D (cid:48) The classical rising factorials is the sequence ( p n ) ∞ n =0 of monic polynomials on R ofbinomial type that are explicitly given by p n ( t ) = ( t ) n := t ( t + 1)( t + 2) · · · ( t + n − . The corresponding delta operator is Q = 1 − e − D , so that Q is the difference operator( Qp )( t ) = p ( t ) − p ( t −
1) for p ∈ P ( R ). The generating function of the rising factorialsis given by ∞ (cid:88) n =0 u n n ! ( t ) n = exp[ − t log(1 − u )] = (1 − u ) − t . t ) n = ( − n ( − t ) n .Let us now consider the corresponding lifted sequence of polynomials, ( P ( n ) ) ∞ n =0 . Wewill call these polynomials the rising factorials on D (cid:48) , and we will write ( ω ) n := P ( n ) ( ω )for ω ∈ D (cid:48) .By (5.8), Q ( x ) = 1 − e − D ( x ) . Hence, by Boole’s formula,( Q ( x ) P )( ω ) = P ( ω ) − P ( ω − δ x ) , x ∈ R d , P ∈ P ( D (cid:48) ) , and by (5.9),( Q ( ζ ) P )( ω ) = (cid:10) ζ ( x ) , P ( ω ) − P ( ω − δ x ) (cid:11) , ζ ∈ D (cid:48) , P ∈ P ( D (cid:48) ) . By (5.11), the generating function is equal to ∞ (cid:88) n =0 n ! (cid:104) ( ω ) n , ξ ⊗ n (cid:105) = exp (cid:2) (cid:104) ω, − log(1 − ξ ) (cid:105) (cid:3) . (5.28) Proposition 5.10.
We have ( ω ) n = ( − n ( − ω ) n for all n ∈ N , and the followingexplicit formulas hold: ( ω ) = 1;( ω ) = ω ;( ω ) n ( x , . . . , x n ) = ω ( x )( ω ( x ) + δ x ( x )) × ( ω ( x ) + δ x ( x ) + δ x ( x )) · · · ( ω ( x n ) + δ x ( x n ) + · · · + δ x n − ( x n )) for n ≥ .Proof. By (5.24), ∞ (cid:88) n =0 n ! (cid:104) ( − n ( − ω ) n , ξ ⊗ n (cid:105) = ∞ (cid:88) n =0 n ! (cid:104) ( − ω ) n , ( − ξ ) ⊗ n (cid:105) = exp (cid:2) (cid:104) ω, − log(1 − ξ ) (cid:105) (cid:3) . Hence, by (5.28), ( ω ) n = ( − n ( − ω ) n for all ω ∈ D (cid:48) and n ∈ N . From this andProposition 5.4, the statement follows. D (cid:48) Let us fix a parameter α ∈ R \ { } . The classical Abel polynomials on R correspondingto the parameter α is the monic polynomial sequence ( p n ) ∞ n =0 of binomial type that hasthe delta operator Q = De αD , i.e., ( Qp )( t ) = p (cid:48) ( t + α ) for p ∈ P ( R ). Thus, Q = q ( D ),where q ( u ) = ue αu . The generating function of the Abel polynomials is given by ∞ (cid:88) n =0 u n n ! p n ( t ) = exp (cid:2) tα − W ( αu ) (cid:3) , W is the inverse function of u (cid:55)→ ue u (around 0), the so-called Lambert W -function.Consider the corresponding lifted sequence of polynomials ( P ( n ) ) ∞ n =0 , the Abel poly-nomials on D (cid:48) . By Lemma 2.20 and (5.8), Q ( x ) = D ( x ) e αD ( x ) = D ( x ) E ( αδ x ) , x ∈ R d , and by (5.9), Q ( ζ ) P = (cid:10) ζ ( x ) , D ( x ) P ( · + αδ x ) (cid:11) , ζ ∈ D (cid:48) , P ∈ P ( D (cid:48) ) . By (5.11), the generating function is equal to ∞ (cid:88) n =0 n ! (cid:104) P ( n ) ( ω ) , ξ ⊗ n (cid:105) = exp (cid:2)(cid:10) ω, α − W ( αξ ) (cid:11)(cid:3) . We have α − W ( αu ) = ∞ (cid:88) k =1 ( − αk ) k − k ! u k . Hence, by Proposition 5.1, (cid:104) P ( n ) ( ω ) , ξ ⊗ n (cid:105) = (cid:88) π ∈ P ( n ) (cid:89) B ∈ π ( − α | B | ) | B |− (cid:104) ω, ξ | B | (cid:105) . D (cid:48) of binomial type Let us recall that the (monic) Laguerre polynomials on R corresponding to a parameter k ≥ −
1, ( p [ k ] n ) ∞ n =0 , have the generating function ∞ (cid:88) n =0 u n n ! p [ k ] n ( t ) = exp (cid:20) tu u (cid:21) (1 + u ) − ( k +1) . (5.29)In particular, for the parameter k = −
1, the Laguerre polynomial sequence ( p n ) ∞ n =0 :=( p [ − n ) ∞ n =0 has the generating function ∞ (cid:88) n =0 u n n ! p n ( t ) = exp (cid:20) tu u (cid:21) . (5.30)Hence, the polynomial sequence ( p n ) ∞ n =0 is of binomial type and its delta operator is Q = q ( D ) with q ( u ) = u − u = ∞ (cid:88) k =1 u k . P ( n ) ) ∞ n =0 will be called the Laguerre polynomialsequence on D (cid:48) of binomial type . Thus, for each x ∈ R d , the delta operator Q ( x ) of( P ( n ) ) ∞ n =0 is given by Q ( x ) = q ( D ( x )) = D ( x )1 − D ( x ) = ∞ (cid:88) k =1 D ( x ) k , and the corresponding generating function is ∞ (cid:88) n =0 n ! (cid:104) P ( n ) ( ω ) , ξ ⊗ n (cid:105) = exp (cid:20)(cid:28) ω, ξ ξ (cid:29)(cid:21) . (5.31)By Proposition 5.1, the polynomial (cid:104) P ( n ) ( ω ) , ξ ⊗ n (cid:105) has representation (5.13) with α k = ( − k +1 k ! . In view of the factor k ! in α k , we can also give the following combi-natorial formula for (cid:104) P ( n ) ( ω ) , ξ ⊗ n (cid:105) .Let W ( n ) denote the collection of all sets β = { b , b , . . . , b k } such that each b i = ( j , . . . , j l i ) is an element of { , , . . . , n } l i with j u (cid:54) = j v if u (cid:54) = v , and each j ∈ { , , . . . , n } is a coordinate of exactly one b i ∈ β . For β ∈ W ( n ) and b i =( j , . . . , j l i ) ∈ β , we denote | b i | := l i .By (5.13), we now get, for the Laguerre polynomials, (cid:104) P ( n ) ( ω ) , ξ ⊗ n (cid:105) = (cid:88) β ∈ W ( n ) (cid:89) b ∈ β (cid:104)− ω, ( − ξ ) | b | (cid:105) . (5.32) Definition . Let ( Q ( ζ )) ζ ∈D (cid:48) be a family of delta operators. We say that a monicpolynomial sequence on D (cid:48) , ( S ( n ) ) ∞ n =0 , is a Sheffer sequence for the family of deltaoperators ( Q ( ζ )) ζ ∈D (cid:48) if for each ζ ∈ D (cid:48) and f ( n ) ∈ D (cid:12) n , n ∈ N , Q ( ζ ) (cid:104) S ( n ) , f ( n ) (cid:105) = (cid:104) S ( n − , A ( ζ ) f ( n ) (cid:105) , (6.1)where A ( ζ ) is the annihilation operator, see Definition 2.17.Of course, any basic sequence for a family of delta operators is a Sheffer sequencefor that family of delta operators. Theorem 6.2.
Let ( Q ( ζ )) ζ ∈D (cid:48) be a family of delta operators and let ( P ( n ) ) ∞ n =0 be itsbasic sequence, which has the generating function (4.2) . Let ( S ( n ) ) ∞ n =0 be a monicpolynomial sequence on D (cid:48) . Then the following conditions are equivalent: (SS1) ( S ( n ) ) ∞ n =0 is a Sheffer sequence for the family ( Q ( ζ )) ζ ∈D (cid:48) . There is a unique operator T ∈ S ( P ( D (cid:48) )) such that, for each f ( n ) ∈ D (cid:12) n , n ∈ N , T (cid:104) S ( n ) , f ( n ) (cid:105) = (cid:104) P ( n ) , f ( n ) (cid:105) . (6.2)(SS3) The sequence ( S ( n ) ) ∞ n =0 has the generating function ∞ (cid:88) n =0 n ! (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) = exp[ (cid:104) ω, A ( ξ ) (cid:105) ] τ ( A ( ξ )) , ω ∈ D (cid:48) , ξ ∈ D , (6.3) where A ( ξ ) ∈ S ( D , D ) is given by (4.3) and τ ∈ S ( D , R ) is such that τ (0) = 1 . (SS4) For each n ∈ N and ω, ζ ∈ D (cid:48) , S ( n ) ( ω + ζ ) = n (cid:88) k =0 (cid:18) nk (cid:19) S ( k ) ( ω ) (cid:12) P ( n − k ) ( ζ ) . (6.4)(SS5) There is a ( ρ ( n ) ) ∞ n =0 ∈ F ( D (cid:48) ) with ρ (0) = 1 such that, for each n ∈ N , S ( n ) ( ω ) = n (cid:88) k =0 (cid:18) nk (cid:19) ρ ( k ) (cid:12) P ( n − k ) ( ω ) , ω ∈ D (cid:48) . Remark . For the meaning of the right hand side of formula (6.3), see Proposition A.1and Remark A.2.
Proof of Theorem 6.2. (SS1) ⇒ (SS2). We define a linear operator T : P ( D (cid:48) ) → P ( D (cid:48) )by formula (6.2). Since ( P ( n ) ) ∞ n =0 and ( S ( n ) ) ∞ n =0 are monic polynomial sequences on D (cid:48) ,we have T ∈ L ( P ( D (cid:48) )), see formulas (3.3) and (3.4). Thus, we only have to prove that T is shift-invariant. For this purpose, fix any G ( k ) ∈ D (cid:48)(cid:12) k , ξ ∈ D , and k ∈ N . By(6.1) and (6.2), we obtain T (cid:10) G ( k ) ( x , . . . , x k ) , Q ( x ) · · · Q ( x k ) (cid:104) S ( n ) , ξ ⊗ n (cid:105) (cid:11) = ( n ) k (cid:104) G ( k ) , ξ ⊗ k (cid:105) T (cid:104) S ( n − k ) , ξ ⊗ ( n − k ) (cid:105) = ( n ) k (cid:104) G ( k ) , ξ ⊗ k (cid:105)(cid:104) P ( n − k ) , ξ ⊗ ( n − k ) (cid:105) = (cid:10) G ( k ) ( x , . . . , x k ) , Q ( x ) · · · Q ( x k ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) (cid:11) = (cid:10) G ( k ) ( x , . . . , x k ) , Q ( x ) · · · Q ( x k ) T (cid:104) S ( n ) , ξ ⊗ n (cid:105) (cid:11) . Therefore, T (cid:104) G ( k ) ( x , . . . , x k ) , Q ( x ) · · · Q ( x k ) (cid:105) = (cid:104) G ( k ) ( x , . . . , x k ) , Q ( x ) · · · Q ( x k ) (cid:105) T. Hence, by Theorem 4.7, T is shift-invariant.35SS2) ⇒ (SS1). By (6.2) with n = 0, we have T T is invertible and T − ∈ S ( P ( D (cid:48) )). By Corollary 4.10, T − commutes witheach Q ( ζ ), ζ ∈ D (cid:48) . Therefore, for each n ∈ N and ξ ∈ D , we get Q ( ζ ) (cid:104) S ( n ) , ξ ⊗ n (cid:105) = Q ( ζ ) T − (cid:104) P ( n ) , ξ ⊗ n (cid:105) = T − Q ( ζ ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) = T − ( n (cid:104) ζ, ξ (cid:105)(cid:104) P ( n − , ξ ⊗ ( n − (cid:105) ) = n (cid:104) ζ, ξ (cid:105)(cid:104) S ( n − , ξ ⊗ ( n − (cid:105) . (SS2) ⇒ (SS3). For a fixed ζ ∈ D (cid:48) , we apply Theorem 4.7 to the shift-invariantoperator E ( ζ ) T − . This gives( E ( ζ ) T − P )( ω ) = ∞ (cid:88) k =0 k ! (cid:10) G ( k ) ( x , . . . , x k ) , ( Q ( x ) · · · Q ( x k ) P )( ω ) (cid:11) , P ∈ P ( D (cid:48) ) , where (cid:104) G ( k ) , f ( k ) (cid:105) = (cid:0) E ( ζ ) T − (cid:104) P ( k ) , f ( k ) (cid:105) (cid:1) (0) = (cid:0) E ( ζ ) (cid:104) S ( k ) , f ( k ) (cid:105) (cid:1) (0) = (cid:104) S ( k ) ( ζ ) , f ( k ) (cid:105) . Thus,( E ( ζ ) T − P )( ω ) = ∞ (cid:88) k =0 k ! (cid:10) S ( k ) ( ζ )( x , . . . , x k ) , ( Q ( x ) · · · Q ( x k ) P )( ω ) (cid:11) , P ∈ P ( D (cid:48) ) . (6.5)For the family of delta operators ( D ( ζ )) ζ ∈D (cid:48) and its basic sequence (monomials),consider the isomorphism J defined in Corollary 4.11. Hence, by (4.26) and (6.5),( J ( E ( ζ ) T − ))( ξ ) = ∞ (cid:88) k =0 k ! (cid:42) S ( k ) ( ζ ) , (cid:32) ∞ (cid:88) n =1 n ! (cid:0) R ,n ξ ⊗ n (cid:1)(cid:33) ⊗ k (cid:43) . (6.6)Furthermore, by (4.22),( J T )( ξ ) = 1 + ∞ (cid:88) k =1 k ! (cid:104) τ ( k ) , ξ ⊗ k (cid:105) =: τ ( ξ ) , (6.7)where (cid:104) τ ( k ) , ξ ⊗ k (cid:105) := (cid:0) T (cid:104)· ⊗ k , ξ ⊗ k (cid:105) (cid:1) (0) , ξ ∈ D , k ∈ N . (6.8)(Note that the first term in (6.7) is indeed equal to 1, because T maps 1 into 1.) By(4.23) and the isomorphism theorem,exp[ (cid:104) ζ, ξ (cid:105) ] = ( J E ( ζ ))( ξ ) = (cid:0) J ( E ( ζ ) T − ) (cid:1) ( ξ )( J T )( ξ ) . (6.9)By Proposition A.1 (see also Remark A.2) and (6.6)–(6.9), we get ∞ (cid:88) k =0 k ! (cid:42) S ( k ) ( ζ ) , (cid:32) ∞ (cid:88) n =1 n ! (cid:0) R ,n ξ ⊗ n (cid:1)(cid:33) ⊗ k (cid:43) = exp[ (cid:104) ζ, ξ (cid:105) ] τ ( ξ ) , (6.10)36ompare with (4.28). We define operators A k ∈ L ( D (cid:12) k , D ) by formula (4.29) for k ≥ A := . Thus, A ( ξ ) = (cid:80) ∞ k =1 A k ξ ⊗ k ∈ S ( D , D ) is the compositional inverse of (cid:80) ∞ n =1 1 n ! R ,n ξ ⊗ n ∈ S ( D , D ). Now formula (6.10) implies (SS3).(SS3) ⇒ (SS2). For the family of delta operators ( D ( ζ )) ζ ∈D (cid:48) and its basic sequence(monomials), we construct the isomorphism J . We define an operator T ∈ S ( P ( D (cid:48) ))by T := J − τ . Since τ (0) = 1, by Proposition A.1 and Corollary 4.11, there exists anoperator S ∈ S ( P ( D (cid:48) )) satisfying ST = T S = . Hence, the operator T is invertibleand T − = S .Since T − /τ (0) = 1, for each f ( n ) ∈ D (cid:12) n , we obtain (cid:0) T − (cid:104)· ⊗ n , f ( n ) (cid:105) (cid:1) ( ω ) = (cid:104) ω ⊗ n , f ( n ) (cid:105) + n − (cid:88) i =0 (cid:104) ω ⊗ i , g ( i ) (cid:105) (6.11)for some g ( i ) ∈ D (cid:12) i , i = 0 , , . . . , n − D (cid:12) n (cid:51) f ( n ) (cid:55)→ ˜ S ( n ) f ( n ) := T − (cid:104) P ( n ) , f ( n ) (cid:105) ∈ P ( n ) ( D (cid:48) ) . (6.12)Since T − ∈ L ( P ( D (cid:48) )) and satisfies (6.11), and the linear operator D (cid:12) n (cid:51) f ( n ) (cid:55)→ (cid:104) P ( n ) , f ( n ) (cid:105) ∈ P ( n ) ( D (cid:48) )is continuous (see Lemma 3.1), we conclude that ˜ S ( n ) ∈ L ( D (cid:12) n , P ( n ) ( D (cid:48) )). Hence, byLemma 3.1 and (6.11), there exists a monic polynomial sequence ( ˜ S ( n ) ) ∞ n =0 that satisfies(˜ S ( n ) f ( n ) )( ω ) = (cid:104) ˜ S ( n ) ( ω ) , f ( n ) (cid:105) . (6.13)By (6.12) and (6.13), we obtain (cid:104) P ( n ) , f ( n ) (cid:105) = T (cid:104) ˜ S ( n ) , f ( n ) (cid:105) . It follows from the proof of the implication (SS2) ⇒ (SS3) that the monic polynomialsequence ( ˜ S ( n ) ) ∞ n =0 has the same generating function as ( S ( n ) ) ∞ n =0 , so they coincide. Butthis implies (6.2).Thus, we have proved that the conditions (SS1), (SS2), and (SS3) are equivalent.(SS2) ⇒ (SS4). For ω, ζ ∈ D (cid:48) , ξ ∈ D , and n ∈ N we have (cid:104) S ( n ) ( ω + ζ ) , ζ ⊗ n (cid:105) = (cid:0) E ( ζ ) (cid:104) S ( n ) , ξ ⊗ n (cid:105) (cid:1) ( ω )= (cid:0) E ( ζ ) T − (cid:104) P ( n ) , ξ ⊗ n (cid:105) (cid:1) ( ω )= (cid:0) T − E ( ζ ) (cid:104) P ( n ) , ξ ⊗ n (cid:105) (cid:1) ( ω )= (cid:0) T − (cid:104) P ( n ) ( · + ζ ) , ξ ⊗ n (cid:105) (cid:1) ( ω )37 n (cid:88) k =0 (cid:18) nk (cid:19)(cid:0) T − (cid:104) P ( k ) , ξ ⊗ k (cid:105) )( ω ) (cid:104) P ( n − k ) ( ζ ) , ξ ⊗ ( n − k ) (cid:105) = n (cid:88) k =0 (cid:18) nk (cid:19) (cid:104) S ( k ) ( ω ) , ξ ⊗ k (cid:105)(cid:104) P ( n − k ) ( ζ ) , ξ ⊗ ( n − k ) (cid:105) . (SS4) ⇒ (SS5). In formula (6.4), swap ω and ζ , set ζ = 0, and denote ρ ( n ) := S ( n ) (0).Note that ρ (0) = S (0) (0) = 1.(SS5) ⇒ (SS1). For each ζ ∈ D (cid:48) , ξ ∈ D , and n ∈ N , we get from (SS5) Q ( ζ ) (cid:104) S ( n ) , ξ ⊗ n (cid:105) = n (cid:88) k =0 (cid:18) nk (cid:19) (cid:104) ρ ( k ) , ξ ⊗ k (cid:105) Q ( ζ ) (cid:104) P ( n − k ) , ξ ⊗ ( n − k ) (cid:105) = n − (cid:88) k =0 (cid:18) nk (cid:19) (cid:104) ρ ( k ) , ξ ⊗ k (cid:105) ( n − k ) (cid:104) ζ, ξ (cid:105)(cid:104) P ( n − k − , ξ ⊗ ( n − k − (cid:105) = n (cid:104) ζ, ξ (cid:105) n − (cid:88) k =0 (cid:18) n − k (cid:19) (cid:104) ρ ( k ) , ξ ⊗ k (cid:105)(cid:104) P ( n − k − , ξ ⊗ ( n − k − (cid:105) = n (cid:104) ζ, ξ (cid:105)(cid:104) S ( n − , ξ ⊗ ( n − (cid:105) . Corollary 6.4.
Let the conditions of Theorem 6.2 be satisfied. Then ( S ( n ) ) ∞ n =0 is aSheffer sequence for the family ( Q ( ζ )) ζ ∈D (cid:48) if and only if the sequence ( S ( n ) ) ∞ n =0 has thegenerating function ∞ (cid:88) n =0 n ! (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) = exp (cid:2) (cid:104) ω, A ( ξ ) (cid:105) − C ( A ( ξ )) (cid:3) , ω ∈ D (cid:48) , ξ ∈ D , (6.14) where A ( ξ ) := (cid:80) ∞ k =1 A k ξ ⊗ k ∈ S ( D , D ) is as in (4.2) and C ( ξ ) := (cid:80) ∞ k =1 (cid:104) C ( k ) , ξ ⊗ k (cid:105) ∈S ( D , R ) is such that C (0) = 0 .Proof. We note that (cid:104) ω, A ( ξ ) (cid:105) ∈ S ( D , R ) is the composition of (cid:104) ω, ξ (cid:105) ∈ S ( D , R ) and A ( ξ ) ∈ S ( D , D ). As easily seen, exp[ (cid:104) ω, A ( ξ ) (cid:105) ] ∈ S ( D , R ) in formula (6.3) can beunderstood as the composition of exp t = (cid:80) ∞ n =0 t n n ! ∈ S ( R , R ) and (cid:104) ω, A ( ξ ) (cid:105) ∈ S ( D , R ),see Definition A.3.Consider log(1 + t ) = (cid:80) ∞ n =1 ( − n +1 t n n ∈ S ( R , R ). Define C ( ξ ) ∈ S ( D , R ) as thecomposition of log(1 + t ) ∈ S ( R , R ) and τ ( ξ ) − ∈ S ( D , R ) (note that τ (0) − C (0) = 0. By Remark A.8, we obtain τ ( ξ ) = exp( C ( ξ )), the equality in S ( D , R ).Hence, using again Remark A.8, we conclude that formula (6.3) can be written as(6.14). Remark . According to Theorem 6.2 and Corollary 6.4, any Sheffer sequence ( S ( n ) ) ∞ n =0 is completely identified by a family of delta operators, ( Q ( ζ )) ζ ∈D (cid:48) , and a formal series C ∈ S ( D , R ) with C (0) = 0. 38 orollary 6.6. Under the conditions of Theorem 6.2, assume that ( S ( n ) ) ∞ n =0 is a Shef-fer sequence. Let ( κ ( n ) ) ∞ n =0 ∈ F ( D (cid:48) ) with κ (0) = 1 satisfy τ ( A ( ξ )) = ∞ (cid:88) k =0 k ! (cid:104) κ ( k ) , ξ ⊗ k (cid:105) . (6.15) Then, for each n ∈ N , P ( n ) ( ω ) = n (cid:88) k =0 (cid:18) nk (cid:19) κ ( k ) (cid:12) S ( n − k ) ( ω ) , ω ∈ D (cid:48) . (6.16) Proof.
By (4.2), (SS3) and (6.15), ∞ (cid:88) n =0 n ! (cid:104) P ( n ) ( ω ) , ξ ⊗ n (cid:105) = (cid:32) ∞ (cid:88) k =0 k ! (cid:104) κ ( k ) , ξ ⊗ k (cid:105) (cid:33) (cid:32) ∞ (cid:88) n =0 n ! (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) (cid:33) , which implies (6.16). Corollary 6.7.
Under the conditions of Theorem 6.2, assume that ( S ( n ) ) ∞ n =0 is a Shef-fer sequence. Then, ( S ( n ) ) ∞ n =0 = ( P ( n ) ) ∞ n =0 if and only if S ( n ) (0) = 0 for each n ∈ N .Proof. By (SS4), for each n ∈ N and ω ∈ D (cid:48) , we have S ( n ) ( ω ) = P ( n ) ( ω ) + n (cid:88) k =1 (cid:18) nk (cid:19) S ( k ) (0) (cid:12) P ( n − k ) ( ω ) . From here the statement follows.Let C ( D (cid:48) ) denote the cylinder σ -algebra on D (cid:48) , i.e., the minimal σ -algebra on D (cid:48) withrespect to which each monomial (cid:104)· , ξ (cid:105) ( ξ ∈ D ) is measurable. A probability measure µ on ( D (cid:48) , C ( D (cid:48) )) is said to have finite moments if, for any n ∈ N and f ( n ) ∈ D (cid:12) n , (cid:90) D (cid:48) |(cid:104) ω ⊗ n , f ( n ) (cid:105)| dµ ( ω ) < ∞ . We now propose the following definition.
Definition . Let µ be a probability measure on ( D (cid:48) , C ( D (cid:48) )) that has finite moments.Let ( S ( n ) ) ∞ n =0 be a polynomial sequence on D (cid:48) . The polynomials ( S ( n ) ) ∞ n =0 are said to be orthogonal with respect to µ if for any m, n ∈ N , m (cid:54) = n , f ( m ) ∈ D (cid:12) m , and g ( n ) ∈ D (cid:12) n , (cid:90) D (cid:48) (cid:104) S ( m ) ( ω ) , f ( m ) (cid:105)(cid:104) S ( n ) ( ω ) , g ( n ) (cid:105) dµ ( ω ) = 0 . orollary 6.9. Let µ be a probability measure on ( D (cid:48) , C ( D (cid:48) )) that has finite moments.Let ( S ( n ) ) ∞ n =0 be a Sheffer sequence on D (cid:48) . Assume that ( S ( n ) ) ∞ n =0 are orthogonal withrespect to µ . Then the corresponding operator T has the following representation: ( T P )( ω ) = (cid:90) D (cid:48) P ( ω + ζ ) dµ ( ζ ) , P ∈ P ( D (cid:48) ) . (6.17) Furthermore, for f ( n ) ∈ D (cid:12) n , n ∈ N , we have (cid:104) τ ( n ) , f ( n ) (cid:105) = (cid:90) D (cid:48) (cid:104) ω ⊗ n , f ( n ) (cid:105) dµ ( ω ) . (6.18) Here, τ ( ξ ) = 1 + (cid:80) ∞ n =1 1 n ! (cid:104) τ ( n ) , ξ ⊗ n (cid:105) ∈ S ( D , R ) is as in (6.3) .Remark . Formula (6.18) states that each τ ( n ) is the n th moment of the orthogo-nality measure µ . Proof of Corollary 6.9.
Note that formula (6.17) holds for P = 1. Now, for each n ∈ N and ξ ∈ D , we obtain by (SS4): (cid:90) D (cid:48) (cid:104) S ( n ) ( ω + ζ ) , ξ ⊗ n (cid:105) dµ ( ζ ) = n (cid:88) k =0 (cid:18) nk (cid:19) (cid:104) P ( n − k ) ( ω ) , ξ ⊗ ( n − k ) (cid:105) (cid:90) D (cid:48) (cid:104) S ( k ) ( ζ ) , ξ ⊗ k (cid:105) dµ ( ζ )= (cid:104) P ( n ) ( ω ) , ξ ⊗ n (cid:105) = (cid:0) T (cid:104) S ( n ) , ξ ⊗ n (cid:105) (cid:1) ( ω ) . Formula (6.18) follows immediately from (6.8) and (6.17).
Remark . Note that, in the proof of Corollary 6.9, we only use the fact that (cid:82) D (cid:48) (cid:104) S ( n ) ( ω ) , f ( n ) (cid:105) dµ ( ω ) = 0 for all n ∈ N . Corollary 6.12.
Assume that the conditions of Corollary 6.9 are satisfied. Assumethat there exists
O ⊂ D , an open neighborhood of zero in D , such that, for all ξ ∈ O , (cid:90) D (cid:48) exp (cid:2) |(cid:104) ω, ξ (cid:105)| (cid:3) dµ ( ω ) < ∞ . Then, for all ξ ∈ O , τ ( ξ ) = (cid:90) D (cid:48) exp (cid:2) (cid:104) ω, ξ (cid:105) (cid:3) dµ ( ω ) , (6.19) i.e., τ ( ξ ) is the Laplace transform of the measure µ .Proof. Formula (6.19) follows from (6.18) and the dominated convergence theorem.Recall that a Sheffer sequence on R whose delta operator is the operator of differ-entiation is called an Appell sequence on R .40 efinition . Let ( S ( n ) ) ∞ n =0 be a Sheffer sequence for the family of delta operators( Q ( ζ )) ζ ∈D (cid:48) = ( D ( ζ )) ζ ∈D (cid:48) . Then we call ( S ( n ) ) ∞ n =0 an Appell sequence on D (cid:48) .By (4.2), A ( ξ ) = ξ in the case of an Appell sequence. Hence, an Appell sequence( S ( n ) ) ∞ n =0 has the generating function ∞ (cid:88) n =0 n ! (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) = exp (cid:2) (cid:104) ω, ξ (cid:105) − C ( ξ ) (cid:3) , ω ∈ D (cid:48) , ξ ∈ D . R We can extend the procedure described in Section 5 to a lifting of Sheffer sequenceson R . Let ( s n ) ∞ n =0 be a Sheffer sequence of monic polynomials on R for the deltaoperator Q , i.e., Qs n = ns n − for each n ∈ N . Thus, Q has representation (5.1) andthe polynomial sequence ( s n ) ∞ n =0 has the generating function ∞ (cid:88) n =0 u n n ! s n ( t ) = exp[ ta ( u ) − c ( a ( u ))] , where a ( u ) ∈ S ( R , R ) is given by (5.3) (being the compositional inverse of the q givenby (5.2)) and c ( u ) = ∞ (cid:88) k =1 c k u k ∈ S ( R , R ) . We now consider the family of delta operators, ( Q ( ζ )) ζ ∈D (cid:48) , given by (5.9). Then A ( ξ ) is given by (5.10). Furthermore, for k ∈ N , we define C ( k ) ∈ D (cid:48)(cid:12) k by (cid:104) C ( k ) , f ( k ) (cid:105) := c k (cid:104) D k f ( k ) (cid:105) , f ( k ) ∈ D (cid:12) k . Here, the operator D k is defined by (5.5) and for ξ ∈ D , we denote (cid:104) ξ (cid:105) := (cid:82) R d ξ ( x ) dx .Thus, for ξ ∈ D , we define C ( ξ ) := ∞ (cid:88) k =1 (cid:104) C ( k ) , ξ ⊗ k (cid:105) = ∞ (cid:88) k =1 c k (cid:104) ξ k (cid:105) =: (cid:104) c ( ξ ) (cid:105) . We now consider the Sheffer sequence ( S ( n ) ) ∞ n =0 for the family of delta operators( Q ( ζ )) ζ ∈D (cid:48) that has the generating function ∞ (cid:88) n =0 n ! (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) = exp (cid:2) (cid:104) ω, a ( ξ ) (cid:105) − (cid:104) c ( a ( ξ )) (cid:105) (cid:3) , ω ∈ D (cid:48) , ξ ∈ D , (7.1)see (5.4) and Corollary 6.4. Thus, we may think of the Sheffer sequence ( S ( n ) ) ∞ n =0 on D (cid:48) as the lifting of the Sheffer sequence ( s n ) ∞ n =0 on R .41 roposition 7.1. Let ( S ( n ) ) ∞ n =0 be a Sheffer sequence with generating function (7.1) .Then, for each n ∈ N , ρ ( n ) := S ( n ) (0) ∈ D (cid:48)(cid:12) n satisfies (cid:104) ρ ( n ) , ξ ⊗ n (cid:105) = (cid:88) π ∈ P ( n ) (cid:89) B ∈ π λ | B | (cid:104) ξ | B | (cid:105) , ξ ∈ D , (7.2) where λ k ∈ R are defined by λ ( u ) := − c ( a ( u )) = ∞ (cid:88) k =1 λ k k ! u k , u ∈ R . (7.3) Furthermore, (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) = n (cid:88) k =0 (cid:18) nk (cid:19) (cid:104) ρ ( k ) , ξ ⊗ k (cid:105)(cid:104) P ( n − k ) ( ω ) , ξ ⊗ ( n − k ) (cid:105) , ω ∈ D (cid:48) , ξ ∈ D , (7.4) where (cid:104) P n ( ω ) , ξ ⊗ n (cid:105) is given by formula (5.13) .Proof. By (7.1) with ω = 0 and (7.3), we have ∞ (cid:88) n =0 n ! (cid:104) ρ ( n ) , ξ ⊗ n (cid:105) = exp (cid:34) ∞ (cid:88) k =1 λ k k ! (cid:104) ξ k (cid:105) (cid:35) . (7.5)Using Fa`a di Bruno’s formula for the n th derivative of composition of functions, wededuce (7.2) from (7.5). Formula (7.4) immediately follows from (SS4) with ω = 0 (or(SS5)), (5.11), and Proposition 5.1.We can also write down the result of Proposition 7.1 in the following form. Bya marked partition of the set { , , . . . , n } we will mean a pair ( π, m π ) in which π = { B , B , . . . , B k } ∈ P ( n ) and m π : π → {− , + } . (The value m π ( B i ) ∈ {− , + } maybe interpreted as the mark of the element B i of the partition π ). We will denote by MP ( n ) the collection of all marked partitions of { , , . . . , n } . Corollary 7.2.
Let ( S ( n ) ) ∞ n =0 be a Sheffer sequence with generating function (7.1) .Then, for each n ∈ N , ω, ∈ D (cid:48) , and ξ ∈ D , (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) = (cid:88) ( π, m π ) ∈ MP ( n ) (cid:89) B ∈ π : m π ( B )=+ α | B | (cid:104) ω, ξ | B | (cid:105) (cid:89) B ∈ π : m π ( B )= − λ | B | (cid:104) ξ | B | (cid:105) , see formula (5.12) for the definition of α k .Proof. Immediate from Proposition 7.1. 42sing Proposition 7.1, we can now immediately extend Corollary 5.2 to the case ofa lifted Sheffer sequence.
Corollary 7.3.
Let ( S ( n ) ) ∞ n =0 be a Sheffer sequence with generating function (7.1) .Then, for each η ∈ M ( R d ) and n ∈ N , we have S ( n ) ( η ) ∈ M sym (( R d ) n ) . Furthermore,for each Λ ∈ B ( R d ) , and n ∈ N , we have (cid:0) S ( n ) ( η ))(Λ n ) = ˜ s n ( η (Λ)) , (7.6) where (˜ s n ) ∞ n =0 is the Sheffer sequence on R with generating function ∞ (cid:88) n =0 u n n ! ˜ s n ( t ) = exp (cid:2) ta ( u ) − vol(Λ) c ( a ( u )) (cid:3) , where vol(Λ) := (cid:82) Λ dx . In particular, (˜ s n ) ∞ n =0 = ( s n ) ∞ n =0 if vol(Λ) = 1 . Proposition 7.4.
The statement of Proposition 5.3 remains true for a Sheffer sequence ( S ( n ) ) ∞ n =0 with generating function (7.1) .Proof. Analogously to the proof of Proposition 5.3, we note that, for any ξ, φ ∈ D satisfying (5.17),exp (cid:2) (cid:104) ω, a ( ξ + φ ) (cid:105) − (cid:104) c ( a ( ξ + φ )) (cid:105) (cid:3) = exp (cid:2) (cid:104) ω, a ( ξ ) (cid:105) − (cid:104) c ( a ( ξ )) (cid:105) (cid:3) exp (cid:2) (cid:104) ω, a ( φ ) (cid:105) − (cid:104) c ( a ( φ )) (cid:105) (cid:3) . The rest of the proof is similar to that of Proposition 5.3.We will now consider examples of lifted Sheffer sequences. D (cid:48) The sequence of the Hermite polynomials on R , ( s n ) ∞ n =0 , is the Appell sequence on R with c ( u ) = u . The Hermite polynomials are orthogonal with respect to the standardGaussian (normal) distribution on R . The lifting of ( s n ) ∞ n =0 is the sequence of Hermitepolynomials on D (cid:48) , ( S ( n ) ) ∞ n =0 , that has the generating function ∞ (cid:88) n =0 n ! (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) = exp (cid:20) (cid:104) ω, ξ (cid:105) − (cid:104) ξ (cid:105) (cid:21) = exp (cid:20) (cid:104) ω, ξ (cid:105) − (cid:107) ξ (cid:107) L ( R d ,dx ) (cid:21) . (7.7) Remark . For each ξ ∈ D with (cid:107) ξ (cid:107) L ( R d ,dx ) = 1, we get from (7.7) that (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) = s n ( (cid:104) ω, ξ (cid:105) ) . (7.8)We see that the Hermite polynomials, S ( n ) , do not actually make use of the spatialstructure of the underlying space, D (cid:48) , but essentially use only the Hilbert space struc-ture of L ( R d , dx ). Formula (7.8) is an exceptional property of the infinite-dimensionalHermite polynomials, compare with the general case discussed in Corollary 7.3.43sing either Proposition 7.1 or Corollary 7.2, we easily get an explicit formula (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) = [ n ] (cid:88) k =0 (cid:18) n k (cid:19) (2 k )! k ! 2 k ( −(cid:104) ξ (cid:105) ) k (cid:104) ω, ξ (cid:105) n − k , where [ n ] denotes the largest integer ≤ n . (Note that (2 k )! k ! 2 k is the number of all partitions π ∈ P (2 k ) such that each set from the partition π has precisely two elements.)Let µ be the probability measure on D (cid:48) that has Fourier transform (cid:90) D (cid:48) exp[ i (cid:104) ω, ξ (cid:105) ] dµ ( ω ) = exp (cid:20) − (cid:107) ξ (cid:107) L ( R d ,dx ) (cid:21) , ξ ∈ D . The measure µ is called the Gaussian white noise measure . The Hermite polynomials( S ( n ) ) ∞ n =0 are orthogonal with respect to µ , and furthermore, for any m, n ∈ N , f ( m ) ∈D (cid:12) m , and g ( n ) ∈ D (cid:12) n , (cid:90) D (cid:48) (cid:104) S ( m ) ( ω ) , f ( m ) (cid:105)(cid:104) S ( n ) ( ω ) , g ( n ) (cid:105) dµ ( ω ) = δ m,n n !( f ( m ) , g ( n ) ) L ( R d ,dx ) (cid:12) n , (7.9)see e.g. [6, 15].As pointed out in the Introduction, the infinite dimensional Hermite polynomialsare well-known and play a fundamental role in Gaussian white noise analysis, see e.g. [6,15, 16, 31] and the references therein. In white noise analysis, one usually writes : ω ⊗ n :for S ( n ) ( ω ) and call it the n th Wick power of ω . In that context, the transformation T given by formula (6.17) is known as the C -transform. D (cid:48) The sequence of the Charlier polynomials on R , ( s n ) ∞ n =0 , is the Sheffer sequence with a ( u ) = log(1 + u ) and c ( u ) = e u −
1, so that λ ( u ) = − u . The Charlier polynomialsare orthogonal with respect to the Poisson distribution corresponding to the intensityparameter 1. The lifting of ( s n ) ∞ n =0 is the sequence of the Charlier polynomials on D (cid:48) ,( S ( n ) ) ∞ n =0 , that has the generating function ∞ (cid:88) n =0 n ! (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) = exp [ (cid:104) ω, log(1 + ξ ) (cid:105) − (cid:104) ξ (cid:105) ] . Note that the corresponding binomial sequence is (( ω ) n ) ∞ n =0 , the falling factorials on D (cid:48) .By Proposition 7.1, (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) = n (cid:88) k =0 (cid:18) nk (cid:19) (cid:104)− ξ (cid:105) k (cid:104) ( ω ) n − k , ξ ⊗ ( n − k ) (cid:105) . (7.10)44urthermore, by Corollary 6.6, we obtain (cid:104) ( ω ) n , ξ ⊗ n (cid:105) = n (cid:88) k =0 (cid:18) nk (cid:19) (cid:104) ξ (cid:105) k (cid:104) S ( n − k ) ( ω ) , ξ ⊗ ( n − k ) (cid:105) . (7.11)Compare formulas (7.10) and (7.11) with Corollaries 2.9 and 2.10 in [20], respectively.Note that the latter results were obtained only for ω from the configuration space Γ.Let µ be the probability measure on D (cid:48) that has Fourier transform (cid:90) D (cid:48) exp[ i (cid:104) ω, ξ (cid:105) ] dµ ( ω ) = exp (cid:20)(cid:90) R d ( e iξ ( x ) − dx (cid:21) , ξ ∈ D . The measure µ is concentrated on the configuration space Γ and is called the Poissonpoint process , or the
Poisson white noise measure . The Charlier polynomials ( S ( n ) ) ∞ n =0 are orthogonal with respect to the Poisson point process µ and formula (7.9) holds truein this case.The Charlier polynomials ( S ( n ) ) ∞ n =0 play a fundamental role in Poisson analysis, seee.g. [17, 19, 22]. In this analysis, the transformation T given by formula (6.17) is alsoknown as the C -transform. D (cid:48) It follows from (5.29) that, for each parameter k > −
1, the sequence of the Laguerrepolynomials ( p [ k ] n ) ∞ n =0 on R corresponding to the parameter k is a Sheffer sequence,whose corresponding binomial sequence is ( p n ) ∞ n =0 = ( p [ − n ) ∞ n =0 , see (5.30). For each k > −
1, the Laguerre polynomials ( p [ k ] n ) ∞ n =0 are orthogonal with respect to the gammadistribution 1Γ( k + 1) χ (0 , ∞ ) ( t ) t k e − t dt. In particular, for k = 0, the Laguerre polynomials ( s n ) ∞ n =0 := ( p [0] n ) ∞ n =0 are orthogonalwith respect to the exponential distribution χ (0 , ∞ ) ( t ) e − t dt on R . By (5.29), ( s n ) ∞ n =0 is the Sheffer sequence with a ( u ) = u u and c ( u ) = − log(1 − u ), so that λ ( u ) = − log(1 + u ).The lifting of ( s n ) ∞ n =0 is the sequence ( S ( n ) ) ∞ n =0 of the Laguerre polynomials on D (cid:48) that has the generating function ∞ (cid:88) n =0 n ! (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) = exp (cid:20)(cid:28) ω, ξ ξ (cid:29) − (cid:104) log(1 + ξ ) (cid:105) (cid:21) . (7.12)Note that the corresponding polynomial sequence of binomial type is the Laguerresequence ( P ( n ) ) ∞ n =0 with generating function (5.31), see Subsection 5.4. Analogously toformula (5.32), we will now present a combinatorial formula for (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) .45e can identify each permutation π ∈ S ( n ) with c ( π ) := { ν , . . . , ν k } , the set ofthe cycles in π . For each cycle ν i ∈ c ( π ), we denote by | ν i | the length of the cycle ν i .We define MS ( n ) := (cid:8) ( π, m π ) | π ∈ S ( n ) , m π : c ( π ) → { + , −} (cid:9) , compare with the definition of MP ( n ) above.Note that, for a given subset of N that has m elements, there are ( m − m that contain the points from this set. Hence, by Corollary 7.2 and (7.12),we get: (cid:104) S ( n ) ( ω ) , ξ ⊗ n (cid:105) = (cid:88) ( π, m π ) ∈ MS ( n ) (cid:89) ν ∈ c ( π ): m π ( ν )=+ | ν | (cid:10) − ω, ( − ξ ) | ν | (cid:11) (cid:89) ν ∈ c ( π ): m π ( ν )= − (cid:10) ( − ξ ) | ν | (cid:11) . By Corollary 7.3, formula (7.6) holds with (˜ s n ) ∞ n =0 = ( p [ k ] n ) ∞ n =0 , the Laguerre poly-nomials on R corresponding to the parameter k = vol(Λ) − > − µ be the probability measure on D (cid:48) that has the Laplace transform (cid:90) D (cid:48) e −(cid:104) ω,ξ (cid:105) dµ ( ω ) = exp (cid:20) − (cid:90) R d log(1 + ξ ( x )) dx (cid:21) , ξ ∈ D , ξ > − . The µ is called the gamma measure , or the gamma completely random measure . It isconcentrated on the set of all (positive) discrete Radon measures η = (cid:80) i s i δ x i ∈ M ( R d )with s i > i . Note that, with µ -probability one, the set of atoms of η , { x i } , isdense in R d .As follows from [21, 22], the Laguerre polynomials ( S ( n ) ) ∞ n =0 are orthogonal withrespect to the gamma measure µ , and furthermore, for any ξ, ψ ∈ D and m, n ∈ N , (cid:90) D (cid:48) (cid:104) S ( m ) ( ω ) , ξ ⊗ m (cid:105)(cid:104) S ( n ) ( ω ) , ψ ⊗ n (cid:105) dµ ( ω ) = δ m,n n ! (cid:88) π ∈ S ( n ) (cid:89) ν ∈ c ( π ) (cid:10) ( ξψ ) | ν | (cid:11) . The Laguerre polynomials ( S ( n ) ) ∞ n =0 play a fundamental role in gamma analysis,see e.g. [21, 22, 25, 26]. Acknowledgments
The authors acknowledge the financial support of the SFB 701 “Spectral structuresand topological methods in mathematics”, Bielefeld University. MJO was supportedby the Portuguese national funds through FCT—Funda¸c˜ao para a Ciˆencia e a Tecnolo-gia, within the project UID/MAT/04561/2013. YK and DF were supported by theEuropean Commission under the project STREVCOMS PIRSES-2013-612669.46 ppendix: Formal tensor power series
We fix a general Gel’fand triple (2.1). The following proposition is a direct consequenceof formula (2.9).
Proposition A.1.
Let F ( ξ ) = (cid:80) ∞ n =0 (cid:104) F ( n ) , ξ ⊗ n (cid:105) ∈ S (Φ , R ) be such that F (0) = F (0) (cid:54) =0 . Then there exists a unique (cid:80) ∞ n =0 (cid:104) G ( n ) , ξ ⊗ n (cid:105) ∈ S (Φ , R ) such that (cid:32) ∞ (cid:88) n =0 (cid:104) F ( n ) , ξ ⊗ n (cid:105) (cid:33) (cid:32) ∞ (cid:88) n =0 (cid:104) G ( n ) , ξ ⊗ n (cid:105) (cid:33) = 1 . Explicitly, G (0) = 1 /F (0) and for n ≥ , G ( n ) is recursively given by G ( n ) = − F (0) n − (cid:88) i =0 F ( n − i ) (cid:12) G ( i ) . We will denote (cid:32) ∞ (cid:88) n =0 (cid:104) F ( n ) , ξ ⊗ n (cid:105) (cid:33) − := ∞ (cid:88) n =0 (cid:104) G ( n ) , ξ ⊗ n (cid:105) . Remark
A.2 . It follows from Proposition A.1 that, for any F ( ξ ) , G ( ξ ) ∈ S (Φ , R ) with F (0) (cid:54) = 0, we obtain G ( ξ ) F ( ξ ) ∈ S (Φ , R ) . Definition
A.3 . Let R ( t ) = (cid:80) ∞ n =0 r n t n ∈ S ( R , R ). For each F ( ξ ) = (cid:80) ∞ n =1 (cid:104) F ( n ) , ξ ⊗ n (cid:105) ∈S (Φ , R ) with F (0) = 0, we define a composition of R and F , denoted by R ◦ F ( ξ ) or R ( F ( ξ )) = ∞ (cid:88) n =0 r n (cid:32) ∞ (cid:88) k =1 (cid:104) F ( k ) , ξ ⊗ k (cid:105) (cid:33) n , as the formal series G ( ξ ) = (cid:80) ∞ n =0 (cid:104) G ( n ) , ξ ⊗ n (cid:105) ∈ S (Φ , R ) with G (0) := r and G ( n ) := n (cid:88) m =1 (cid:88) ( k ,...,k m ) ∈ N m k + ··· + k m = n r m (cid:0) F ( k ) (cid:12) F ( k ) (cid:12) · · · (cid:12) F ( k m ) (cid:1) , n ∈ N . Definition
A.4 . Let A ( ξ ) = (cid:80) ∞ n =1 A n ξ ⊗ n , B ( ξ ) = (cid:80) ∞ n =1 B n ξ ⊗ n ∈ S (Φ , Φ). We define a composition of A and B , denoted by A ◦ B ( ξ ) or A ( B ( ξ )) = ∞ (cid:88) n =1 A n (cid:32) ∞ (cid:88) k =1 B k ξ ⊗ k (cid:33) ⊗ n ,
47s the formal series C ( ξ ) = (cid:80) ∞ n =1 C n ξ ⊗ n ∈ S (Φ , Φ) with C n := n (cid:88) m =1 (cid:88) ( k ,...,k m ) ∈ N m k + ··· + k m = n A m (cid:0) B k (cid:12) B k (cid:12) · · · (cid:12) B k m (cid:1) , n ∈ N . (A.1)Here, for ( k , . . . , k m ) ∈ N m with k + · · · + k m = n , we denote B k (cid:12) B k (cid:12) · · · (cid:12) B k m := Sym n ( B k ⊗ B k ⊗ · · · ⊗ B k m ) , where Sym n ∈ L (Φ ⊗ n , Φ (cid:12) n ) is the operator of symmetrization, see (2.2).Similarly to Definitions A.3 and A.4, we give the following Definition
A.5 . Let F ( ξ ) = (cid:80) ∞ n =0 (cid:104) F ( n ) , ξ ⊗ n (cid:105) ∈ S (Φ , R ) and A ( ξ ) = (cid:80) ∞ n =1 A n ξ ⊗ n ∈S (Φ , Φ). We define a composition of F and A , denoted by F ◦ A ( ξ ) or F ( A ( ξ )) = ∞ (cid:88) n =0 (cid:42) F ( n ) , (cid:32) ∞ (cid:88) k =1 A k ξ ⊗ k (cid:33) ⊗ n (cid:43) , as the formal series G ( ξ ) = (cid:80) ∞ n =0 (cid:104) G ( n ) , ξ ⊗ n (cid:105) ∈ S (Φ , R ) with G (0) := F (0) and G ( n ) := n (cid:88) m =1 (cid:88) ( k ,...,k m ) ∈ N m k + ··· + k m = n ( A ∗ k (cid:12) A ∗ k (cid:12) · · · (cid:12) A ∗ k m ) F ( m ) , n ∈ N . Here A ∗ k ∈ L (Φ (cid:48) , Φ (cid:48)(cid:12) k ) is the adjoint of A k . Remark
A.6 . It follows from Definition A.5 that F ( A ( ξ )) = F (0) + ∞ (cid:88) n =1 n (cid:88) m =1 (cid:88) ( k ,...,k m ) ∈ N m k + ··· + k m = n (cid:10) F ( m ) , ( A k ξ ⊗ k ) (cid:12) ( A k ξ ⊗ k ) (cid:12) · · · (cid:12) ( A k m ξ ⊗ k m ) (cid:11) . Proposition A.7.
Let F ( ξ ) ∈ S (Φ , R ) and let A ( ξ ) , B ( ξ ) ∈ S (Φ , Φ) . Then ( F ◦ A ) ◦ B ( ξ ) = F ◦ ( A ◦ B )( ξ ) , the equality in S (Φ , R ) .Proof. The proposition follows from Definitions A.4 and A.5, see also Remark A.6. Weleave the details to the interested reader. 48 emark
A.8 . In view of Proposition A.7, we may just write F ◦ A ◦ B ( ξ ). As easilyseen, a similar statement also holds for the composition S ◦ R ◦ F ( ξ ) ∈ S (Φ , R ),where S, R ∈ S ( R , R ) and F ∈ S (Φ , R ), and for the composition A ◦ B ◦ C ( ξ ), where A, B, C ∈ S (Φ , Φ).
Definition
A.9 . Let A ( ξ ) ∈ S (Φ , Φ). Then B ( ξ ) ∈ S (Φ , Φ) is called the compositionalinverse of A ( ξ ) if A ◦ B ( ξ ) = B ◦ A ( ξ ) = ξ . Remark
A.10 . Note that, if B ( ξ ) ∈ S (Φ , Φ) is the compositional inverse of A ( ξ ) ∈S (Φ , Φ), then A ( ξ ) is the compositional inverse of B ( ξ ). Proposition A.11.
Let A ( ξ ) = (cid:80) ∞ n =1 A n ξ ⊗ n ∈ S (Φ , Φ) with A ∈ L (Φ) being ahomeomorphism. Then there exists a unique compositional inverse B ( ξ ) of A ( ξ ) .Proof. Let us first prove that there exists a unique B ( ξ ) := (cid:80) ∞ n =1 B n ξ ⊗ n ∈ S (Φ , Φ)such that A ◦ B ( ξ ) = ξ . It follows from formula (A.1) that C = A B . Hence, for C = , we must have B = A − . Now, by (A.1) for n ≥
2, we get C n = A B n + n (cid:88) m =2 (cid:88) ( k ,...,k m ) ∈ N m k + ··· + k m = n A m (cid:0) B k (cid:12) B k (cid:12) · · · (cid:12) B k m (cid:1) . Hence, we get C n =0 for n ≥ B n = − A − n (cid:88) m =2 (cid:88) ( k ,...,k m ) ∈ N m k + ··· + k m = n A m (cid:0) B k (cid:12) B k (cid:12) · · · (cid:12) B k m (cid:1) . Similarly, we prove that there exists a unique ˜ B ( ξ ) := (cid:80) ∞ n =1 ˜ B n ξ ⊗ n ∈ S (Φ , Φ) suchthat ˜ B ◦ A ( ξ ) = ξ . Here ˜ B = A − and for n ≥ B n = − n − (cid:88) m =1 (cid:88) ( k ,...,k m ) ∈ N m k + ··· + k m = n ˜ B m (cid:0) A k (cid:12) A k (cid:12) · · · (cid:12) A k m (cid:1) ( A − ) ⊗ n . Finally, we prove that B ( ξ ) = ˜ B ( ξ ). Indeed, we get, using Remark A.8,˜ B ( ξ ) = ˜ B ◦ ( A ◦ B )( ξ ) = ( ˜ B ◦ A ) ◦ B ( ξ ) = B ( ξ ) . Hence, the proposition is proven.
Remark
A.12 . It follows from Proposition A.11 and its proof that, if A ( ξ ) = (cid:80) ∞ n =1 A n ξ ⊗ n ∈S (Φ , Φ) with A = , then its compositional inverse B ( ξ ) := (cid:80) ∞ n =1 B n ξ ⊗ n exists and B = . 49 eferences [1] Albeverio, S., Daletsky, Y.L., Kondratiev, Y.G., Streit, L.: Non-Gaussian infinite-dimensional analysis. J. Funct. Anal. 138 (1996), 311–350.[2] Anshelevich, M.: Appell polynomials and their relatives. Int. Math. Res. Not.2004, no. 65, 3469–3531.[3] Anshelevich, M.: Appell polynomials and their relatives. II. Boolean theory. Indi-ana Univ. Math. J. 58 (2009), 929–968.[4] Anshelevich, M.: Appell polynomials and their relatives. III. Conditionally freetheory. Illinois J. Math. 53 (2009), 39–66.[5] Barnabei, M., Brini, A., Nicoletti, G.: A general umbral calculus in infinitelymany variables. Adv. Math. 50 (1983), 49–93.[6] Berezansky, Y.M., Kondratiev, Y.G.: Spectral methods in infinite-dimensionalanalysis. Vol. 1, 2. Kluwer Academic Publishers, Dordrecht, 1995 (translated fromthe 1988 Russian original).[7] Berezansky, Y.M., Kondratiev, Y.G., Kuna, T., Lytvynov, E.: On a spectralrepresentation for correlation measures in configuration space analysis. MethodsFunct. Anal. Topology 5 (1999), no. 4, 87–100.[8] Berezansky, Y.M., Sheftel, Z.G., Us, G.F.: Functional analysis. Vol. II. Birkh¨auser,Basel, 1996.[9] Bourbaki, N.: Topological vector spaces. Chapters 1–5. Springer-Verlag, Berlin,1987.[10] Brown, J.: On multivariable Sheffer sequences. J. Math. Anal. Appl. 69 (1979),398–410.[11] Casalis, M.: The 2 d + 4 simple quadratic natural exponential families on R dd