A unified algebraic underpinning for the Hahn polynomials and rational functions
aa r X i v : . [ m a t h . C A ] S e p A unified algebraic underpinning for the Hahnpolynomials and rational functions
Luc Vinet ∗ , Alexei Zhedanov † Centre de Recherches Math´ematiques, Universit´e de Montr´eal,P.O. Box 6128, Centre-ville Station, Montr´eal (Qu´ebec), H3C 3J7, Canada. School of Mathematics, Renmin University of China, Beijing, 100872, China
September 15, 2020
Abstract
An algebra denoted m H with three generators is introduced and shown to admit embeddingsof the Hahn algebra and the rational Hahn algebra. It has a real version of the deformedJordan plane as a subalgebra whose connection with Hahn polynomials is established. Repre-sentation bases corresponding to eigenvalue or generalized eigenvalue problems involving thegenerators are considered. Overlaps between these bases are shown to be bispectral orthogonalpolynomials or biorthogonal rational functions thereby providing a unified description of thesefunctions based on m H . Models in terms of differential and difference operators are used toidentify explicitly the underlying special functions as Hahn polynomials and rational functionsand to determine their characterizations. An embedding of m H in U ( sl ) is presented. A Pad´eapproximation table for the binomial function is obtained as a by-product. This paper introduces an algebra that subsumes the Hahn algebra [1], [2] and the one recentlyidentified [3], which encodes the bispectral properties of the rational functions of Hahn type. Assuch it offers a unified algebraic interpretation of both the Hahn polynomials [4] and rationalfunctions. This algebra has three generators and derives from a cubic potential [5]. It standsto prove fundamental.Put simply bispectral problems are situations where functions satisfy a pair of (generalized)eigenvalue equations in the variable and the spectral parameter. Their study was initiatedsystematically in [6] and they have wide-ranging connections and applications [7] in particularin time and band limiting [8]. With their recurrence relations and differential or differenceequations, the hypergeometric orthogonal polynomials [4] are a particular instance of suchproblems. ∗ E-mail: [email protected] † E-mail: [email protected] Q n ( x, α, β, N ) defined by Q n ( x, α, β, N ) = F (cid:18) − n, n + α + β + 1 , − xα + 1 , − N ; 1 (cid:19) , n = 0 , , . . . N. (1.1)The Hahn algebra H which is attached to these polynomials and their duals has two generators K , K verifying the relations:[ K , [ K , K ]] = aK + bK + c K + d I (1.2)[ K , [ K , K ]] = a { K , K } + bK + c K + d I . (1.3)As usual [ A, B ] = AB − BA and { A, B } = AB + BA . Here a, b, c , c , d , d are taken asreal parameters. Allowing for affine transformations of the generators their number can bereduced to two. When H is realized by the bispectral operators of the polynomials, theyare given in terms of the parameters α and β of the polynomials. It is known (e.g. from theliterature already cited) that the overlaps between eigenvectors of K and K in representationsof dimension N + 1 are given in terms of Hahn polynomials.Biorthogonal rational functions (BRFs) are known to arise as solutions of GeneralizedEigenvalue Problems (GEVPs) involving two tridiagonal matrices [13]. It is also understoodthat some of these BRFs [14], [15], [16], [17], [18], [19] are bispectral in that they are solutionsof two GEVPs. A first foray into the algebraic interpretation of the bispectrality of BRFs wasrecently achieved in [3] by looking at the rational functions of Hahn type: U n ( x ; α, β, N ) = ( − n ( − N ) n ( β + 1) n F (cid:18) − x, − n, β + n − N − N, α − x ; 1 (cid:19) , n = 0 , , . . . N. (1.4)(We use the standard notation for Pochhammer symbols and generalized hypergeometric series[4].) This led to the extension of Leonard pairs to Leonard pencils corresponding to the twoGEVPs defining the bispectral framework. These two pencils were found to be composed outof three operators X, Y, Z which in the case of the rational functions U n given above were foundto satisfy the rational Hahn algebra r H with the following defining relations:[ Z, X ] = Z + Z (1.5)[ X, Y ] = ξ ( X + Z ) + { X, Z } + { Y, Z } + ξ X + ξ Z + Y + ξ I (1.6)[ Y, Z ] = 3 X + Z + ξ { X, Z } + ξ X + ξ Z + ξ I . (1.7)The parameters ξ i , i = 0 , , , , α and β in the realization in terms of the operators that definethe bispectral problem of which the rational functions U n are solutions.We shall explain below that both the Hahn algebra H and the rational Hahn algebra r H canbe embedded in an algebra with three generators { X, Z, V } which we will call the meta-Hahnalgebra m H and describe in the next section. This is the basis of the joint account of the twobispectral problems. When dealing with a linear pencil, P = A + σB , one can set as was done2n [20], [21] the two-parameter problem P u = τ u . Depending on whether σ or τ is taken asthe eigenvalue, one is either looking at a GEVP involving the operators A − τ and B or atan ordinary eigenvalue problem (EVP) for the operator P . This is how the treatment of theHahn polynomials and rational functions will be unified and how m H will be seen to providea synthesizing algebraic framework for these two families of special functions.The exposition will proceed as follows. In addition to presenting m H in Section 2 andproviding its Casimir element, we shall indicate that it is built as an enlargement of thedeformed Jordan plane. How the Hahn algebra H and the rational Hahn algebra r H injectinto the meta-Hahn algebra m H will be described in Section 3. Certain elements of the m H representation theory will be developed in Section 4. We shall consider bases correspondingto the GEVP defined by the generators X and Z and to the adjoint GEVP, and also basescorresponding to the EVPs associated to V and to the pencil X + µZ as well as to theiradjoints. The bispectral properties of various overlaps between these bases will be establishedin Section 5. A differential realization of m H will be given in Section 6 and used on the onehand to obtain the explicit expressions of the bispectral overlaps and on the other to deriveproperties of these special functions. The biorthogonal Hahn rational functions and the Hahnorthogonal polynomials will be recovered there. The embedding of m H in U ( su (2)) is the topicof Section 7. A summary and an outlook will serve as conclusion in Section 8. Two appendicescomplete the paper. Appendix A will provide computational details on the orthogonality ofthe eigenvectors of V and of its adjoint V T in the differential model of m H . Appendix B looksat Jacobi polynomials with a parameter that is a negative integer. It will offer the restrictedversions of Gauss’ and Pfaff’s transformation formulas that are valid in this case and that wereused in Section 6. It will furthermore include a Pad´e approximation table for the binomialfunction as an interesting by-product. m H We start with the algebra that has two generators X and Z verifying[ Z, X ] = Z + Z. (2.1)We will comment below on this noteworthy algebra, see Remark 2.2. We now extend it by theaddition of one extra generator V and a central element I . We assume that the commutationrelations of V with X and Z minimally depart from those of Lie algebras by exhibiting a singlequadratic term and that they are of the form[ X, V ] = η { V, Z } + η X + η V + η Z + η I , (2.2)[ V, Z ] = η X + η Z + η I , (2.3)where η , . . . η are some (real) parameters. Enforcing the Jacobi relation[ V, [ Z, X ]] + [ Z, [ X, V ]] + [ X, [ V, Z ]] = 0 (2.4)leads to:
Proposition 2.1
The relations (2.1) - (2.2) are compatible with the Jacobi identity (2.4) ifand only if the constraints η = η = 1 , η = η (2.5) are imposed.
3t is appropriate to fix a standardized form of the emerging algebra so as to identify how manyessential parameters it contains. This number is reduced to three with the help of the followingautomorphisms: V → κV X → X + ζZ (2.6)that preserve the commutation relation between X and Z . We shall not consider here thedegenerate case η = 0; under this proviso it is possible to choose κ so that η = 2 and to pick ζ in order to have η = − η . This brings us to the following: Definition 2.1
The meta-Hahn algebra m H is generated by the three elements V , X and Z that satisfy the relations [ Z, X ] = Z + Z, (2.7)[ X, V ] = { V, Z } + η X + V − η Z + η I , (2.8)[ V, Z ] = 2 X + η Z + η I . (2.9)It is further observed that m H derives from a potential. Given F f = C [ x , x , . . . x n ] afree associative algebra with n generators, F cycl = F f / [ F f , F f ]. F cycl has the cyclic words[ x i x i . . . x i r ] as basis. The cyclic derivative ∂∂x j : F cycl → F f is such that ∂ [ x i x i . . . x i r ] ∂x j = X { s | i s = j } x i s +1 x i s +2 . . . x i r x i x i . . . x i s − (2.10)and is extended to F cycl by linearity. Let Φ( x , . . . x n ) ∈ F cycl . An algebra whose definingrelations are given by ∂ Φ ∂x j = 0 , j = 1 , . . . n (2.11)is said to derive from the potential Φ. Now let x = V , x = X and x = Z and takeΦ = [ XV Z ] − [ XZV ] − [ V Z ] − [ V Z ] − [ X ] + η Z ] − η [ XZ ] − η [ X ] − η [ Z ] (2.12)It is readily seen that the relations (2.7), (2.8), and (2.9) of m H are given by ∂ Φ ∂V = 0, ∂ Φ ∂Z = 0, ∂ Φ ∂X = 0 respectively.This fact points at the existence of a non-trivial central element with terms correspondingto those in Φ with the exclusion of [ XV Z ] − [ XZV ]. Proposition 2.2
The Casimir element of m H is given by: Q = { V, Z + Z } + 2 X + (2 − η ) Z + η { X, Z } + 2 η X + 2( η + 1) Z. (2.13)Let us record that before the automorphisms (2.6) are used to set η = − η and η = 2 theelement Q such that [ Q, X ] = [
Q, Z ] = [
Q, V ] = 0 has the following expression: Q = { V, Z + Z } + η X + η { X, Z } + ( η + η ) Z + 2 η X + (2 η + η ) Z (2.14)which is of course consistent with (2.13). Remark 2.1
We shall at times allow for η and η to be in fact central. emark 2.2 The meta-Hahn algebra has a two-dimensional subalgebra generated by X and Z with commutation relation given by (2.7) . Its remarkable connection to Hahn polynomialswill be stressed in Sections 4 and 6. If we consider this algebra over the field of complexnumbers, upon setting X = i ˆ X and Z = i ˆ Z − , the commutation relation is converted to [ ˆ Z, ˆ X ] = − ˆ Z − . This is recognized as the deformed Jordan plane denoted J in [22]. Thisalgebra is known [23] to be a PBW deformation of the two-dimensional Artin-Schelter regularalgebra [ X, Z ] = Z , which is called the Jordan plane or meromorphic Weyl algebra and hasbeen much studied, see for instance [24], [25], [26]. (We would like to also cite [27] for thestudy of the representations of the algebras defined by the relation [ X, Z ] = h ( Z ) with h apolynomial in Z .) One main feature of m H is that it contains the algebras H and r H ; this explains the prefixmeta. We now describe the two embeddings. H ֒ → m H Introduce the element W = X + µZ (3.1)where µ is an arbitrary real parameter. Using the commutation relations (2.7), (2.8) and (2.9)of m H , it is checked that upon taking K = W and K = V, (3.2)the relations (1.2) and (1.3) of the Hahn algebra H are satisfied with a = 2 , b = 2 µ − η + 2 η , c = − , d = − Q, (3.3) c = − η ( η + 2) , d = η (2 µ − η ) − η η (1 + µ ) . (3.4)The following proposition thus follows: Proposition 3.1
The centrally extended Hahn algebra admits an embedding in the meta-Hahnalgebra with V and W as generators and with the parameters of (1.2) and (1.3) given by (3.4) . The proof is straigthforward. Note that the Casimir element Q of m H given in (2.13) occursin the parameter d ; to be precise, it is hence a central extension of the Hahn algebra that isthus injected in the meta-Hahn algebra.Observe that the commutation relations between the generators V, W, Z form a closed setthat reads: [
Z, W ] = Z + Z, (3.5)[ W, V ] = { V, Z } + τ W + V + τ Z + τ I , (3.6)[ V, Z ] = 2 W + τ Z + τ I , (3.7)with τ = η − µη , τ = η − µ, τ = 2 µ − η µ − η , τ = η . (3.8)5 emark 3.1 These relations (3.5) , (3.6) , (3.7) between { V, W, Z } have the same form asthose between { V, X, Z } that define m H and are given in (2.7) , (2.8) , (2.9) . They actuallyhave an extra parameter in front of Z in the commutation relation between V and W ; thissimply follows from the fact that the pencil parameter µ in W reintroduces the free constantthat we had taken away in the definition of m H by using precisely this freedom (see (2.6) ) inthe definition of X . r H ֒ → m H Instead of using a linear combination of generators as in the preceding subsection, the embed-ding of the rational Hahn algebra in the meta one is achieved by supplementing X and Z withthe generator Y constructed as the following product: Y = XV. (3.9)We are then led to:
Proposition 3.2
A central extension of the rational Hahn algebra is embedded in the meta-Hahn algebra by taking
X, Y = XV, Z as generators. The coefficients ξ i , i = 0 , . . . , enteringin the relations (1.5) , (1.6) and (1.7) of r H are given as follows in terms of the parameters η , η , η of m H : ξ = 12 ( η − Q ) , ξ = η + 1 , ξ = η + η + 1 , (3.10) ξ = η + η + η + 1 , ξ = 2 η + 1 . (3.11)The proof is again straightforward and simply relies on the repeated use of the commutationrelations (2.7), (2.8) and (2.9) of m H . Note that here also the Casimir element Q given in(2.13) must be called upon to rexpress terms such as V Z and
V Z in the generators. Thisexplains how Q arises in ξ and why it is a central extension of r H that is embedded in m H . m H This section will develop certain aspects of the representation theory of the meta-Hahn algebra.We shall leave to another occasion its systematic study and will here be content with bringingto light the features that are most relevant to the bispectrality issues. We shall be concernedwith finite dimensional representations on a real vector space V N of dimension N + 1 withscalar product denoted by ( , ).We shall introduce a number of bases:1. The basis { d n , n = 0 , . . . , N } for the solution space of the GEVP Xd n = λ n Zd n ; (4.1)2. The basis { d ∗ n , n = 0 , . . . , N } associated to the adjoint GEVP X T d ∗ n = µ n Z T d ∗ n ; (4.2)3. The eigenbasis { e n , n = 0 , . . . , N } of V , solutions of the EVP V e n = ν n e n ; (4.3)6. The eigenbasis { f n , n = 0 , . . . , N } of the linear pencil X + µZ , with( X + µZ ) f n = ρ n f n . (4.4)We shall also consider the adjoint bases { e ∗ n } and { f ∗ n } such that V T e ∗ n = νe ∗ n and ( X T + µZ T ) f ∗ n = ρ n f ∗ n . We shall determine the various spectra and the shapes of the matricesrepresenting the generators in these bases.The sets of eigenvalues { λ n } and { µ n } coincide and are given by the roots of the charac-teristic polynomial of degree N + 1 given by the determinant | X − λZ | .It is readily seen that the sets { d n } and { ˜ d n = Z T d ∗ n } form two biorthogonal families ofvectors: ( d m , ˜ d n ) = 0 , m = n with ˜ d n = Z T d ∗ n . (4.5)Indeed we have:( Xd m , d ∗ m ) = λ m ( Zd m , d ∗ n ) = ( d m , X T d ∗ n ) = λ n ( d m , Z T d ∗ n ) = λ n ( Zd m , d ∗ n ); (4.6)and hence ( λ m − λ n )( Zd m , d ∗ n ) = 0 which implies (4.5) assuming λ m = λ n . { d n } We shall now obtain features of the matrices representing X , Z and V in the basis given bythe vectors satisfying the GEVP Xd n = λ n Zd n . Proposition 4.1
The eigenvalues λ n of the GEVP Xd n = λ n Zd n are linear and of the form λ n = α − n, n = 0 , . . . N with α ∈ R . (4.7) Z is a raising operator that acts as follows in the basis { d n } : Zd n = a n d n +1 − d n , Zd N = − d N . (4.8) where a n is a n -dependent factor that depends on the normalization of the basis vectors. Proof. With the help of the commutation relation (2.7), it is straightforward to obtain theintertwining relation( X − ˜ λ n Z )( Z + 1) = ( Z + 1)( X − λ n Z ) with ˜ λ n = λ n − . (4.9)It follows that with d n solution of (4.1), ( Z + 1) d n will be solution of the same GEVP witheigenvalue ˜ λ n = λ n −
1. The eigenvalues are thus incremented by unit steps and hence dependlinearly on n . We shall take Z + 1 to be a raising operator; ( Z T + 1) will therefore be lowering.This leads to the formula (4.7) for the eigenvalues and allows to take the action of Z on d n asgiven by (4.8). This is a rare example of a GEVP that can be solved algebraically. Proposition 4.2
The generator V is represented in the basis { d n } by a lower Hessenbergmatrix, that is, a matrix with zero entries above the first superdiagonal. Proof. Write generically
V d n = P Nm =0 V ( d ) m,n d m . We shall use the constraints that relation (2.9)imposes on the representations. Recall (4.8). Given that X has the same matrix form as Z inview of the GEVP (4.1), acting on d n with the right-hand side of (2.9) will give an expression7ike κ n d n +1 + ζ n d n where κ n and ζ n are some coefficients. The left-hand side will bring [ V, Z ] d n and the entries of the matrix [ V, Z ] are given by[
V, Z ] ( d ) m,n = a n V ( d ) m,n +1 − a m − V ( d ) m − ,n . (4.10)For any matrix A , it is always assumed that A m,n = 0 whenever m or n are either smaller than0 or larger than N .Consider first the entries above the diagonal. Since these are all zero on the right handside, those of [ V, Z ] must vanish. It is readily seen that this implies that all the entries of V above the superdiagonal are 0: V ( d ) n,n + k +1 = 0 , k = 1 , . . . , N − n − . (4.11)Indeed from [ V, Z ] ( d )1 ,k +1 = a k +1 V ( d )1 ,k +2 = 0 for k = 1 , . . . , N −
2, we find that (4.11) is satisfiedfor n = 1. Iterating over n yields the desired result.The entries on the diagonal and subdiagonal are subjected to more involved relations thatwould need to be solved were we to fully obtain the representation. Turn now attention tothe entries below the subdiagonal. Again, since they have no match on the right of (2.9), thecorresponding entries of [ V, Z ] must vanish, which requires some of the matrix elements V ( d ) m,n of V to satisfy a n V ( d ) m,n +1 − a m − V ( d ) m − ,n = 0 , for m > n + 1 , m = n + 2 , . . . , N. (4.12)This confirms that when acting on the vectors { d n } , the only non-zero matrix elements of V are above the superdiagonal.The representations of the algebra m H in the GEVP basis have another striking featurethat we discuss next. Proposition 4.3
In the basis { d n } , the product V ( d ) X ( d ) of the lower Hessenberg matrix V ( d ) and the lower two-diagonal matrix X ( d ) is tridiagonal. The proof of this proposition will make use this time of the relation (2.8) of m H . Recall that Xd n = ( α − n ) Zd n , with Zd n given by (4.8). It is straightforward to derive the followingformulas for the matrix elements of [ X, V ] and { V, Z } :[ X, V ] ( d ) mn = ( α − m + 1) a m − V ( d ) m − ,n + ( m − n ) V ( d ) m,n − ( α − n ) V ( d ) m,n +1 , (4.13) { V, Z } ( d ) m,n = a n V ( d ) m,n +1 − V ( d ) m,n + a m − V ( d ) m − ,n . (4.14)When considering the entries outside the diagonal and the subdiagonal when the left and theright hand sides of (2.8) are applied to d n , only the terms [ X, V ], { V, Z } and V contribute andlead to the following relations between matrix elements of V :( α − m ) a m − V ( d ) m − ,n + ( m − n + 1) V ( d ) m,n − ( α − n + 1) a n V ( d ) m,n +1 = 0 , m = { n, n + 1 } . (4.15)Consider now the matrix elements of the product V X . It is easily seen that(
V X ) ( d ) m,n = ( α − n )( a n V ( d ) m,n +1 − V ( d ) m,n ) . (4.16)Owing to (4.11), above the superdiagonal, that is for m < n −
1, we directly see that (
V X ) m,n =0. For the entries below the subdiagonal, that is when m > n + 1, we first observe that the8onditions (4.12) which we obtained from relation (2.9) do apply. With their help, we see thatequation (4.15) which is a consequence of the defining relation (2.8) can be transformed into( m − n + 1)( a n V ( d ) m,n +1 − V ( d ) m,n ) = 0 , (4.17)as long as m > n + 1. Comparing with the expression (4.16) for ( V X ) ( d ) m,n , this allows toconclude that the matrix elements of V X in the basis { d n } are also zero below the subdiagonaland therefore that remarkably VX is tridiagonal in the GEVP basis. { d ∗ n } This basis is defined by X T d ∗ n = ( α − n ) Z T d ∗ n . The transpose of (4.9) shows that Z T d ∗ n = − d ∗ n + a ( ∗ ) n d ∗ n − (4.18)where a ( ∗ ) n are some other n -dependent constants that are again dependent on the normalizationof the basis vectors. This means that the only non-zero matrix elements of Z in the basis { d ∗ n } are Z ( ∗ ) n +1 ,n = a ( ∗ ) n , Z ( ∗ ) n,n = − . (4.19)Let D be the diagonal operator Dd ∗ n = ( α − n ) d ∗ n . (4.20)(Note that we also have Dd n = ( α − n ) d n .) In the basis { d ∗ n } , we have X T d ∗ n = Z T Dd ∗ n . Thisimplies that the relation between X and Z in that basis is X = DZ and we thus have: Proposition 4.4
The action of X on the vectors d ∗ n is: Xd ∗ n = DZd ∗ n (4.21)= X ( d ∗ ) n +1 ,n d ∗ n +1 + X ( d ∗ ) n,n d ∗ n = ( α − n − a ( ∗ ) n d ∗ n +1 − ( α − n ) d ∗ n . The proof is immediate.The representation of V has the property: Proposition 4.5
The action of V in the basis { d ∗ n } , V d ∗ n = P Nm =0 V ( d ∗ ) m,n d ∗ m , is given by alower Hessenberg matrix V ( d ∗ ) . The proof follows the same lines as that of Proposition 4.2. The defining relation (2.8) imposes V ( d ∗ ) n,n + k +1 = 0 , k = 1 , . . . , N − n − , (4.22)and the conditions a ( ∗ ) n V ( d ∗ ) m,n +1 − a ( ∗ ) m − V ( d ∗ ) m − ,n = 0 , for m > n + 1 , m = n + 2 , . . . , N. (4.23)In addition, we find that in this basis, the analog of Proposition 4.3 has the order of theoperators X and V exchanged. Proposition 4.6
In the basis { d ∗ n } , the product X ( d ∗ ) V ( d ∗ ) is tridiagonal. X, V ] ( d ∗ ) m,n = ( α − m ) a ( ∗ ) m − V ( d ∗ ) m − ,n + ( m − n ) V ( d ∗ ) m,n − ( α − n − V ( d ∗ ) m,n +1 . (4.24)The matrix elements { V, Z } ( d ∗ ) m,n take the same form as (4.14) with V ( d ) replaced by V ( d ∗ ) . For m = { n, n + 1 } , following (2.8),[ X, V ] ( d ∗ ) m,n − { V, Z } ( d ∗ ) m,n − V ( d ∗ ) m,n = (4.25)( α − m − a ( ∗ ) m − V ( d ∗ ) m − ,n + ( m − n + 1) V ( d ∗ ) m,n − ( α − n ) a ( ∗ ) n V ( d ∗ ) m,n +1 = 0 . Now observe that ( XV ) ( d ∗ ) m,n = ( α − n )( a m − V ( d ∗ ) m − ,n − V ( d ∗ ) m,n ) . (4.26)Above the superdiagonal, for m < n + 1, this expression vanishes in view of (4.22). Below thesubdiagonal, for m > n + 1, with the help of (4.23), condition (4.25) becomes( n − m − a ( ∗ ) m − V ( d ∗ ) m − ,n − V ( d ∗ ) m,n ) = 0 , (4.27)from where we see that the entries of ( XV ) ( d ∗ ) below the subdiagonal are also zero, therebyshowing that XV is tridiagonal in the basis { d ∗ n } . V We consider now the eigenbasis { e n } defined by (4.3). Let Xe n = P Nm =0 X ( e ) mn e m and Ze n = P Nm =0 Z ( e ) mn e m . Relevant observations regarding the representations of the meta-Hahn algebrain this basis are summarized here: Proposition 4.7
The eigenvalues ν n of V are quadratic and given by ν n = − n ( n − η − . (4.28) Moreover, X and Z act tridiagonally in the eigenbasis { e n } of V : Xe n = X ( e ) n +1 ,n e n +1 + X ( e ) n,n e n + X ( e ) n − ,n e n − , X ( e ) − , = 0 , X ( e ) N +1 ,N = 0; (4.29) Ze n = Z ( e ) n +1 ,n e n +1 + Z ( e ) n,n e n + Z ( e ) n − ,n e n − , Z ( e ) − , = 0 , Z ( e ) N +1 ,N = 0 . (4.30)Proof. The relations (2.8) and (2.9) of m H respectively impose the following conditions on thematrix elements of X ( e ) and Z ( e ) :( ν n − ν m − η ) X ( e ) m,n = ( ρ n + ν m − η ) Z ( e ) m,n + ( ν n + η ) δ mn , (4.31)( ν m + ν n − η ) Z ( e ) m,n = 2 X ( e ) m,n + η δ mn . (4.32)Assume m = n . Expressing X ( e ) m,n in terms of Z ( e ) m,n , using (4.32) and substituting in (4.31), wesee that the compatibility of these two equations requires that:( ν m − ν n ) + 2( ν m + ν n ) − η ( η + 2) = 0 , m = n. (4.33)For a given n , this quadratic relations allows for two possible values of m which we will taketo be m = n ± X and Z will be10ridiagonal as per (4.29) and (4.30). Upon solving (4.33) we readily see that the eigenvalues ν n are given by (4.28) (the second solution amounts to translating n by η + 1).The representations of m H deserve a detailed study that will be carried out elsewhere. Ontheir own, the representations of the subalgebra consisting of the deformed Jordan plane J generated by X and Z are already bound to be quite rich. It is striking that when the generator V is added to enlarge J to form m H , the diagonalization of V leads to representations of J by tridiadiagonal matrices akin to those examined in [21] in the case of the q -oscillator algebra.One such representation of J has already been obtained in [3]. We shall now view η and η as central charges and take them to have in this representation the following expressions interms of two parameters α, β and N (where N + 1 is the dimension): η = ( N − − β ) α + β + 1 , η = N − − β ; (4.34)we shall also parametrize η in terms of α and β by η = 2 α − β − . (4.35)It can be checked directly that the relations (2.7), (2.8) and (2.9) of m H are satisfied when Vis diagonal, V n,n = − n ( n − N + β ) , (4.36)and X and Z are represented by ( N + 1) × ( N + 1) tridiagonal matrices with non-zero entries: X n +1 ,n = n ( β + n + 1)( N − β − n )( N − β − n )( N − β − n − , (4.37) X n,n = − α − n + βn ( n −
1) + n ( n − N − β − n + 1 − βn ( n + 1) + n ( n + 1) N − β − n − , (4.38) X n − ,n = n ( N − β − n )( N − n + 1)( N − β − n )( N − β − n + 1) , (4.39) Z n +1 ,n = − ( β + n + 1)( N − β − n )( N − β − n )( N − β − n − , (4.40) Z n,n = ( n + 1) β + ( n + 1) N − β − n − − βn + n N − β − n + 1 , (4.41) Z n − ,n = − n ( N − n + 1)( N − β − n )( N − β − n + 1) . (4.42)The range of n is the integers { , , . . . , N } and it is understood that X N +1 ,N = Z N +1 ,N = 0.The Casimir Q of m H given by (2.13) then becomes Q = − α ( α − β ) + 2 α − β − . (4.43)Quite remarkably this representation of the meta-Hahn algebra that can be inferred fromthe representation found in [3] of the rational Hahn algebra allows to connect the deformedJordan plane or the subalgebra generated by X, Z to the Hahn polynomials. Recall [4] thatthe recurrence relation − xQ n ( x ) = A n Q n +1 ( x ) − ( A n + C n ) Q n ( x ) + C n Q n − ( x ) (4.44)11f the Hahn polynomials Q n ( x, ˆ α, ˆ β ) whose explicit expressions are given in (1.1), has forcoefficients A n = ( n + ˆ α + ˆ β + 1)( n + ˆ α + 1)( N − n )(2 n + ˆ α + ˆ β + 1)(2 n + ˆ α + ˆ β + 2) , (4.45) C n = n ( n + ˆ α + ˆ β + N + 1)( n + ˆ β )(2 n + ˆ α + ˆ β )(2 n + ˆ α + ˆ β + 1) . (4.46) Proposition 4.8
The Jacobi matrix ( X + µZ ) m,n obtained from (4.37) - (4.42) is diagonalizedby the orthogonal polynomials ˜ Q n ( y ) = σ n Q n ( x, ˆ α, ˆ β, N ) where Q n ( x ) are the Hahn polynomialsand x = y − α − µ, (4.47)ˆ α = − − µ, (4.48)ˆ β = β + µ − N, (4.49) and σ n = ( − n ( − N ) n ( ˆ α + ˆ β + N + 1) n . (4.50)The proof is straighforward. Allowing for the normalization factors σ n , the following relationsare readily verified using formulas (4.47) to (4.50):( X n +1 ,n + µZ n +1 ,n ) σ n +1 σ n = − A n , (4.51)( X n,n + µZ n,n ) = − α − µ + ( A n + C n ) , (4.52)( X n − ,n + µZ n − ,n ) σ n − σ n = − C n , (4.53)with A n and C n given by (4.45) and (4.46). The conclusion of the proposition then followsfrom comparison with the recurrence relation (4.44). This connection between the deformedJordan plane and the Hahn polynomials takes us to the next subsection. X + µZ We shall now briefly discuss how the generators X , Z and V act on the vectors f n that satisfy( X + µZ ) f n = ρ n f n . We have shown in Section 3 that the pencil W = X + µZ together with V defines an embedding of the Hahn algebra into m H . From the general theory [12], it followsthat when represented on the finite-dimensional space V N , V and W form a Leonard pair. Asa result, V will act tridiagonally on the eigenvectors f n of W . We already know that X and Z separately act in a tridiagonal fashion in the eigenbasis of V ; this will a fortiori be true forthe pencil W as should be from the Leonard perspective.From the observation made in the last subsection that W can be represented (essentially)by the Jacobi matrix of the Hahn polynomials we know that its spectrum will be linear. Thiscan be found directly from the algebra [28]: Proposition 4.9
The spectrum of W is given by ρ n = an + b (4.54) with a and b as affine parameters. W f n = ρ n f n and we know that V acts tridiagonally: V f n = V ( f ) n +1 ,n f n +1 + V ( f ) n,n f n + V ( f ) n − ,n f n − . W and V satisfy the relations of the Hahn algebra(1.2) and (1.3). The last one reads:[ V, [ W, V ]] = a { V, W } + bV + c W + d I . Acting with bothsides on f n , the coefficient of f n +2 yields the condition: ρ n +2 − ρ n +1 + ρ n = 0 , (4.55)which has (4.54) as solution. Remark 4.1
The parameters a and b can obviously be modified by an affine transformation of W . In the representation given in (4.37) - (4.42) , we have as observed a = 1 and b = − α − µ . Remark 4.2
It should be noted that the commutation relations of m H show that this algebradoes not admit representations where the three generators X, Z, V are Hermitian (a fortiorisymmetric) matrices. This is readily seen from the contradictions that follow by trying to im-pose such a constraint. It is manifest however that the Hahn algebra H defined by the relations (1.2) , (1.3) which are verified by W and V is compatible with Hermiticity. Under proper pos-itivity conditions, it is known that H admits unitary, in fact orthogonal, finite-dimensionalrepresentations. In these circumstances W and V can be symmetrized by diagonal similaritytransformations and there is then no distinctions between the EVP and the adjoint EVP bases. Remark 4.3
The action of Z (and of X ) on the eigenvectors { f n } of W can be obtained fromthose of W and V from the relations (3.5) , (3.6) and (3.6) of the (non-standardized) meta-Hahn algebra that these operators verify together with Z . These actions are non-local, that isthey typically correspond to full matrices. This is readily seen. From (3.6) we have { V, Z } + τ Z = [ W, V ] − τ W − V − τ I . (4.56)Using (3.7), we may transform this equation into(2 V − τ − τ ) Z = [ W, V ] + (2 − τ ) W − V + ( τ − τ ) I . (4.57)Note that the right-hand side (rhs) of (4.57) is an algebraic Heun operator [8] and is tridiagonalin both the eigenbases of V and W [29]. Clearly, Z can then be obtained by multiplying thisoperator on the rhs by the inverse of the factor of Z on the left, that is by (2 V − τ − τ ) − .The action of X is then given via X = W − µZ . We are now ready to discuss the bispectral properties of the functions defined by overlapsof the form ( g m , h n ) , m, n = 0 , . . . , N where g m and h m are some of the basis vectors in V N considered in the last section. Recall that the scalar product is taken to be real; to be clear, itwill be defined as follows using row and column vectors:( g m , h n ) = g Tm h n = h Tn g m = ( h n , g m ) . (5.1)The transpose of an operator R on V N is such that( g m , Rh n ) = ( R T g m , h n ) (5.2)13hich is consistent with g Tm Rh n = ( R T g m ) T h n . The overlap ( g m , h n ) involves two discreteindices m and n that will play the role of degree and variable (or vice-versa for the dualfunctions).A key result obtained in [13] states that solutions to GEVPs involving two tridiagonalmatrices are biorthogonal rational functions. This is somehow the analog to Favard’s theorem[30] connecting EVPs for Jacobi matrices to orthogonal polynomials. The overlaps stemmingfrom the representations of the meta-Hahn algebra will lead to functions of those two kinds.The issue will be to recognize first that these characterizations apply and second, that thealgebra m H entails bispectrality. While more space will be dedicated to the less understoodbiorthogonal rational functions, we wish to stress that the meta-Hahn algebra underpins in aunified way the bispectral properties of these two classes of functions. Consider the functions defined as follows as overlaps between the GEVP bases and the eigen-basis of V (and V T ): U m ( n ) = ( e m , d ∗ n ) and ˜ U m ( n ) = ( e ∗ m , Zd n ) . (5.3)where the vectors e ∗ n are the eigenvectors of the transpose of V : V T e ∗ n = ν n e ∗ n . Before we showthat these are rational functions and identify their bispectral properties, let us first observethat they are biorthogonal. We have ( d n , Z T d ∗ k ) = ( Zd n , d ∗ k ) = w − n δ nk , k, n = 0 , . . . , N (5.4)and hence N X n =0 ( u, Zd n )( d ∗ n , v ) w n = ( u, v ) . (5.5)It thus follows that N X n =0 ˜ U k ( n ) U m ( n ) w m = N X n =0 ( e ∗ k , Zd n )( d ∗ n , e m ) w n = ( e ∗ k , e m ) = 0 , k = m. (5.6) U m ( n )From the definition of U m ( n ) and the fact that the vectors d ∗ n satisfy X T d ∗ n = λ n Z T d ∗ n , wehave ( e m , ( X T − λ n Z T ) d ∗ n ) = 0 (5.7)from where it follows that (( X − λ n Z ) e m , d ∗ n ) = 0 . (5.8)Recalling that X and Z have tridiagonal actions on the eigenvectors of V and that the GEVPeigenvalue is λ n = α − n , we find the following result: Proposition 5.1
The function U m ( n ) satisfies the following recurrence relation of GEVP form X ( e ) m +1 ,m U m +1 ( n ) + X ( e ) m,m U m ( n ) + X ( e ) m − ,m U m − ( n ) =( a − n ) h Z ( e ) m +1 ,m U m +1 ( n ) + Z ( e ) m,m U m ( n ) + Z ( e ) m − ,m U m − ( n ) i . (5.9)14et us now identify the other equation of the bispectral system. In a similar fashion wehave (( V − ν m ) e m , d ∗ n ) = 0 (5.10)since e n is an eigenvector of V . Multiplying the first factor by X allows to write the identity( X ( V − ν m ) e m , d ∗ n ) = 0 which leads to( e m , ( V T X T − ν m X T ) d ∗ n ) = 0 . (5.11)Since Y = XV is tridiagonal in the GEVP basis { d ∗ n } , the same will be true for the transposedmatrix Y T = V T X T . We recall that ν m = − m ( m − η −
1) and that (see 4.4) X T d ∗ n = ( α − n )[ − d ∗ n + a ( ∗ ) n d ∗ n − ] . (5.12)This leads to to the following observations: Proposition 5.2
The function U m ( n ) = ( e m , d ∗ n ) is a biorthogonal rational function that sat-isfies the difference equation of GEVP type: Y n,n +1 U m ( n + 1) + Y n,n U m ( n ) + Y n,n − U m ( n −
1) = − m ( m − η − α − n ) h ( α − n )( − U m ( n ) + a ( ∗ ) n U m ( n − i . (5.13)It was already proved to be biorthogonal. That it is a rational function results from the factthat it verifies the above difference GEVP involving two tridiagonal matrices (one being in facttwo-diagonal). These observations are summarized in: Proposition 5.3
The rational function U m ( n ) is bispectral since it satisfies the recurrencerelation (5.9) and the difference equation (5.13) . These properties are algebraically encoded inthe meta-Hahn algebra. ˜ U m ( n )We may describe in the same spirit the bispectrality of the functions ˜ U m ( n ) = ( e ∗ m , Zd n ). Firstobserve that the vectors ˜ d n = Zd n satisfy the GEVP( ˜ X − λ n ˜ Z ) d n = 0 , (5.14)with ˜ X = X + Z + 1 , ˜ Z = Z. (5.15)This simply follows from [ Z, X ] = Z + Z . Let ˜ V = V . The generators ˜ X , ˜ Z and ˜ V verifythe relations (3.5), (3.6), (3.7) with W , Z and V replaced respectively by ˜ X , ˜ Z and ˜ V and theparameters τ given by (differently from (3.8)): τ = 2 η − τ = 2 − η , τ = 2 + η − η − η , τ = η −
2. So apart from the fact that the parameters are different, the structure of therelations obeyed by ˜ X , ˜ Z and ˜ V is the same as that of the relations between X , Z and V . Weobserve that in the new notation:˜ U m ( n ) = (˜ e ∗ n , ˜ d n ) with ˜ V T ˜ e ∗ n = ν n ˜ e ∗ n . (5.16)The essential features of the representations in the bases { ˜ d n } and { ˜ e ∗ n } of the algebra generatedby ˜ X , ˜ Z and ˜ V will hence be the same as those of the meta-Hahn algebra in the bases { d n } and { e ∗ n } ; namely, ˜ X and ˜ Z will be lower bidiagonal in the basis { ˜ d n } and tridiagonal in the15asis { e ∗ n } , V will be lower Hessenberg and V X will be tridiagonal in the basis { ˜ d n } . Thisis confirmed by going again through the arguments that led to the propositions in Subsection4.1 and 4.3. Indeed, these only rely on the structure of the commutation relations whichhas not been modified by going to the tilded generators. Of course, the actual entries of thematrices representing the different operators ˜ X , ˜ Z and ˜ V will be affected by the changes inthe parameters.These observations make clear that the biorthogonal partners ˜ U m ( n ) are also as expected,bispectral rational functions whose properties are determined by an algebra which is essentially m H . As a result, (( ˜ X T − λ n ˜ Z T )˜ e ∗ m , ˜ d n ) = 0 (5.17)leads to their recurrence relation while(˜ e ∗ m , ( ˜ V ˜ X − ˜ ν m X ) ˜ d n ) = 0 (5.18)amounts to their difference equation. The last functions we wish to consider are given by the overlaps S m ( n ) = ( f ∗ n , e m ) and ˜ S m ( n ) = ( f n , e ∗ m ) (5.19)between the eigenvectors of W T = X T + µZ T and those of V and between reciprocally theeigenvectors of W = X + µZ and of V T . With( e ∗ m , e n ) = κ − n δ m,n , ( f ∗ m , f n ) = ζ − n δ m,n , (5.20)and N X n =0 ( u, f n )( f ∗ n , v ) ζ n = ( u, v ) , N X m =0 ( u, e m )( e ∗ m , v ) η n = ( u, v ) , (5.21)we have the relations N X n =0 ˜ S m ( n ) S k ( n ) ζ n = N X n =0 ( f n , e ∗ m )( f ∗ n , e k ) ζ n = κ − m δ m,k , (5.22) N X m =0 ˜ S m ( n ) S m ( k ) κ n = N X m =0 ( f n , e ∗ m )( f ∗ k , e m ) κ m = ζ − n δ n,k . (5.23)Given that V and W verify the commutation relations of the Hahn algebra, within the appro-priate range of the parameters, we expect the functions S m ( n ) = ( f ∗ n , e m ) and ˜ S m ( n ) = ( f n , e ∗ m )to be both related to the same (dual) Hahn polynomials so that (5.22) and (5.23) are orthog-onality relations for a single family of polynomials. To convince oneself that S m ( n ) = ( f ∗ n , e m )and ˜ S m ( n ) = ( f n , e ∗ m ) can be proportional to these polynomials, one shall recall Remark 4.2 asa reminder that the Hahn algebra generators can be symmetrized thus removing the differencebetween the starred and the non-starred bases and rendering S m ( n ) and ˜ S m ( n ) identical. Wewould nevertheless like for generality and practical reasons to keep the adjoint bases and stillconclude that S m ( n ) and ˜ S m ( n ) are expressed in terms of the same polynomials. An argumentto that effect goes like this. From ρ n ( f ∗ n , e m ) = ( W T f ∗ n , e m ) = ( f ∗ n , W e m ) we have ρ n S m ( n ) = W ( e ) m +1 ,m S m +1 ( n ) + W ( e ) m,m S m ( n ) + W ( e ) m − ,m S m − ( n ) . (5.24)16e may thus write S m ( n ) = S ( n ) P m ( n ) with P ( n ) = 1 , (5.25)and conclude from Favard’s theorem that since they obey a three-term recurrence relation, thefunctions P m ( n ) are orthogonal polynomials if W ( e ) m +1 ,m W ( e ) m − ,m > W are real. According to (5.22), the functions ˜ S m ( n ) are themselvesorthogonal (with weight factors) to these polynomials and we can hence conclude that wemust have similarly to (5.25) ˜ S m ( n ) = h ( m ) ˜ S ( n ) P m ( n ) since two systems { P m } , { Q m } , m =0 , . . . , N of orthogonal polynomials that are mutually orthogonal must be related by P m = h ( m ) Q m with h ( m ) a non-zero function of m . This is the desired conclusion, namely, the coreof the functions S m ( n ) and ˜ S m ( n ) is a single family of orthogonal polynomials. The identity(( W − ρ n ) f n , e ∗ m ) = ( f n , ( W T − ρ n ) e ∗ m ) = 0 leads to ρ n ˜ S m ( n ) = W ( e ∗ ) T m +1 ,m ˜ S m +1 ( n ) + W ( e ∗ ) T m,m ˜ S m ( n ) + W ( e ∗ ) T m − ,m ˜ S m − ( n ) (5.27)which must be equivalent to (5.24) and hence implies that h ( k ) W ( e ∗ ) T k,m = W ( e ) k,m .The dual identities ( f ∗ n , ( V − ν m ) e m ) = (( V T − ν m ) f ∗ n , e m ) = 0 and ( f n , ( V T − ν m ) e ∗ m ) =(( V − ν m ) f n , e ∗ m ) = 0 entail the bispectrality of S m ( n ) and ˜ S m ( n ) and the same differenceequation for the polynomials P m ( n ) with the implication that S ( n ) V ( f ∗ ) T k,n = ˜ S ( m ) V ( f ) k,n . Theequation for S m ( n ) is: ν m S m ( n ) = V ( f ∗ ) T n +1 ,n S m ( n + 1) + V ( f ∗ ) T n,n S m ( n ) + V ( f ∗ ) T n − ,n S m ( n − . (5.28)We note that (5.22) and (5.23) embody the orthogonality relations of the polynomials P m ( n )and their duals. These observations can be summed up as follows. Proposition 5.4
The functions S m ( n ) = ( f ∗ n , e m ) are bispectral and obey the three-term re-currence and difference relations (5.24) and (5.28) ; they are proportional to orthogonal poly-nomials. The functions ˜ S m ( n ) = ( f n , e ∗ m ) are also expressed in terms of the same polynomials. Stressing the following point might be appropriate as we conclude this section. To discussspecial functions associated to GEVPs, the adjoint basis is required. This basis is differentfrom that of the original problem. For consistency, we have also introduced the adjoint basesof the EVPs. A reader familiar with Leonard pairs knows that given such a set, say (
V, W ),orthogonal polynomials arise through expanding the vectors of one eigenbasis in terms of thevectors of the other eigenbasis. Here the direct and the adjoint eigenbases can be relatedthrough symmetrizing factors. Expressions such as ( f ∗ n , e m ) provide (up to the normalizationfactors) the expansion coefficients of the vector f n over the basis formed by the vectors e m , m =0 , . . . , N . Concrete examples of such computations will be provided in the next section. We shall now provide realizations of the elements of representation theory developed so farand in this way identify explicitly the special functions whose properties are rooted in m H .17 .1 A differential realization Consider the following operators (we shall use the same symbols for the abstract generatorsand their representations): Z = ( x − I , (6.1) X = x (1 − x ) ddx − α I , (6.2) V = x (1 − x ) d dx + [ x ( N − − β ) − N ] ddx . (6.3)It is readily seen that they obey the commutation relations (2.7), (2.8), (2.9) of m H withthe parameters assigned as in (4.34) and (4.35). Given this representation, the correspondingmodel for the basis { d n } is straightforwardly obtained. It is easily verified that the functions d n ( x ; α ) = γ n x n (1 − x ) − α , n = 0 , , . . . N, (6.4)with γ n a normalization factor, satisfy the differential GEVP Xd n ( x ; α ) = ( α − n ) Zd n ( x ; α ).It is a matter of computation to obtain the actions of Z , X and V on these basis vectors. Onereadily sees that Zd n ( x ; α ) = γ n ( α ) γ n +1 ( α ) d n +1 ( x ; α ) − d n ( x ; α ) = − γ n ( α ) γ n ( α + 1) d n ( x ; α + 1) (6.5)which is in line with (4.8) with a n = γ n ( α ) γ n +1 ( α ) . One also finds Xd n ( x ; α ) = ( α − n ) γ n ( α ) γ n +1 ( α ) d n +1 ( x ; α ) + ( n − α ) d n ( x ; α ) (6.6)= ( n − α ) γ n ( α ) γ n ( α + 1) d n ( x ; α + 1) (6.7)from where one confirms that Xd n ( x ; α ) = ( α − n ) Zd n ( x ; α ). We shall not record the actionof V on d n ( x ; α ) as we shall give below that of V T on d ∗ n ( x ; α ) which has similar features.The adjoint bases will be connected with the formal Lagrange adjoints of the above opera-tors and we will keep the practice of apposing an asterix to the corresponding solutions of theadjoint GEVPs or EVPs. We have Z T = ( x − I , (6.8) X T = − x (1 − x ) ddx + (2 x − α − I , (6.9) V T = x (1 − x ) d dx + [ N + 2 − x ( N + 3 − β )] ddx − ( N + β − I . (6.10)The solutions of ( X T − ( α − n ) Z T ) d ∗ n ( x ; α ) = 0 are recognized to be: d ∗ n ( x ; α ) = γ ∗ n ( α ) x − − n (1 − x ) − α , n = 0 , . . . N, (6.11)where γ ∗ ( α ) are the normalizing coefficients of these basis vectors. The actions of Z T and X T on d ∗ n ( x ; α ) are in this case also simple to derive and one gets: Z T d ∗ n ( x ; α ) = − d ∗ n ( x ; α ) + γ ∗ n ( α ) γ ∗ n − ( α ) d ∗ n − ( x ; α ) = − γ ∗ n ( α ) γ ∗ n ( α + 1) d ∗ n ( x ; α + 1) , (6.12)18 T d ∗ n ( x ; α ) = ( n − α ) d ∗ n ( x ; α ) + ( α − n ) γ ∗ n ( α ) γ ∗ n − ( α ) d ∗ n − ( x ; α ) = ( n − α ) γ ∗ n ( α ) γ ∗ n ( α + 1) d ∗ n ( x ; α + 1) , (6.13)which fits with (4.18) and (4.21) and sets a ∗ n = γ ∗ n ( α ) γ ∗ n − ( α ) . (6.14)Finding the action of V T in this adjoint basis requires a little more work but is also straight-forward and one gets: V T d ∗ n ( x ; α ) = − ( N − n )( n + 1) γ n ( α ) γ n +1 ( α ) d ∗ n +1 ( x ; α )+ [( N − n )( n − α + β + 2) + N ( α − β − d ∗ n ( x ; α )+ ( α − β − α − (cid:20) γ n γ n − d ∗ n − ( x ; α ) + γ n γ n − d ∗ n − ( x ; α ) + · · · + γ n γ d ∗ ( x ; α ) (cid:21) . (6.15)In conformity with Proposition 4.5, we see that V T is a upper Hessenberg matrix. We cantrace the origin of the terms extending upwards beyond the diagonal to the presence of afactor (1 − x ) − = 1 + x + x + . . . which yields the lowering sequence d ∗ n , d ∗ n − , . . . , d ∗ whenit multiplies d ∗ n . Note that truncation must be enforced when d ∗ is reached.The scalar product ( f ( x ) , g ( x )) for f ( x ) , g ( x ) ∈ V N is realized in this model by integratingover a closed contour in the complex plane. Let Γ be a circle | x | = a <
1. With the functionstaken to be on C , we define( f ( x ) , g ( x )) = 12 πi I Γ f ( x ) g ( x ) dx, f, g ∈ V N . (6.16)It is indeed readily seen using Cauchy’s theorem that( d ∗ m , Zd n ) = γ ∗ m γ n πi I Γ x ( n − m − dx = γ ∗ m γ m δ m,n . (6.17)We shall remark that we had introduced generically the normalization factor w − n in ( d ∗ k , Zd n ) = w − n δ kn (see (5.4)), we thus have the identification w − n = γ ∗ n γ n . (6.18)The bases { e m } and { e ∗ m } are realized as follows in this model. The solutions of the EVP V e n ( x ; β, N ) = ν n e n ( x ; β, N ), where ν n = − n ( n − N + β ), are given by the polynomials: e m ( x ; β, N ) = δ m F (cid:18) − m, m − N + β − N ; x (cid:19) , m = 0 , , . . . , N. (6.19)Note that these are Jacobi polynomials where the parameter α is a negative integer and thatthe usual orthogonality relation is not valid since the corresponding integral is not defined.(See [31] for a discussion of Jacobi polynomials with general parameters.) These functions arerather orthogonal under the scalar product given in (6.16) to a set of Laurent polynomialsthat will form the adjoint basis { e ∗ m } . Indeed the EVP V T e ∗ m ( x ; β ) = ν n e ∗ m ( x ; β ) admits thefollowing solutions e ∗ m ( x ; β, N ) = δ ∗ m x − − N F (cid:18) m − N, − m − β − N ; x (cid:19) , m = 0 , , . . . , N, (6.20)19here δ ∗ m are normalization constants. That they form the basis which is such that( e ∗ m , e n ) = δ ∗ m δ n πi I Γ dx x − − N F (cid:18) m − N, − m − β − N ; x (cid:19) F (cid:18) − n, n − N + β − N ; x (cid:19) = ξ n δ m,n (6.21)with ξ n some constants is shown in Appendix A. We shall now identify explicitly the rational functions whose bispectrality is encoded in m H and indicate how their characterization largely follows from this algebraic framework. U m ( n )We shall use the differential realization to determine the functions U m ( n ) = ( e m , d ∗ n ). In themodel described in Section 6.1, U m ( n ) = 12 πi I Γ dx γ ∗ n δ m x − − n (1 − x ) − α F (cid:18) − n, n − N + β − N ; x (cid:19) . (6.22)We hence need to find the residue or the coefficient of x − in the Laurent series of the integrandor equivalently, the coefficient of x n in the expansion:(1 − x ) α − F (cid:18) − n, n − N + β − N ; x (cid:19) = 1 δ m X ℓ γ ∗ n U m ( ℓ ) x ℓ (6.23)which only involves non-negative powers and where we have formally extended the domain of U m . Using the binomial formula (1 − x ) α = ∞ X k =0 ( − α ) k x k k ! , (6.24)one readily obtains U m ( n ) = δ m γ ∗ n (1 − α ) n n ! F (cid:18) − n, − m, m − N + β − N, α − n ; 1 (cid:19) . (6.25)With n as the variable (instead of x ), this is seen to be, up to a normalization factor, therational Hahn function U m ( n ; α, β, N ) defined in (1.4) multiplied by δ m γ ∗ n (1 − α ) n n ! A natural choice for the normalization constants for the vectors d ∗ n ( x ; α ) is γ ∗ n = n !(1 − α ) n (6.26)in which case U m ( n ) = δ m F (cid:16) − n, − m, m − N + β − N, α − n ; 1 (cid:17) . We then find γ ∗ ( α ) γ ∗ n − k ( α ) = n !( n − k )! (1 − α ) n − k (1 − α ) n = ( − n ) k ( α − n ) k (6.27)20ith the help of the standard relations( − n ) k = ( − k n !( n − k )! and ( a ) n − k = ( − k ( a ) n (1 − n − a ) k . (6.28)Define the action of Z , X , and V on U m ( n ) by O U m ( n ) = ( e m , O T d ∗ n ) (6.29)for O any of Z , X , or V . Using the notation T ± f ( n ) = f ( n ±
1) and I f ( n ) = f ( n ), from (6.12),(6.13) and (6.15), we obtain the following realization of the meta-Hahn algebra generators interms of difference operators: Z = −I + nn − α T − , (6.30) X = ( n − α ) I − nT − , (6.31) V = − ( N − n )( n − α + 1) T + + [( N − n )( n − α + β + 2) + N ( α − β − I + ( α − β − α − N X k =1 ( − n ) k ( α − n ) k ( T − ) k . (6.32)It is understood that ( T − ) k U m ( n ) = 0 for k > n. (6.33)Note that in this model X = ( α − n ) Z. (6.34)It is then further observed still in the framework of this realization, that Y = XV = A ( n ) T + + A ( n ) T − + A ( n ) , (6.35)where A ( n ) =( n − α )( n − N )( n + 1 − α ) ,A ( n ) = n ( n − α )( n − α + β − N ) ,A ( n ) =( n − α )( − n + (2 α − N − β ) n − N ( α − . (6.36)This is the translation of the fact observed abstractly that the product V T X T acts in a tridi-agonal fashion in the basis { d ∗ n } . We thus recover the difference realization introduced in [3]as the starting point to the analysis of the bispectral properties of the rational Hahn func-tions (1.4). The basis vectors which obey the GEVP Xf ( n ) = ( α − n ) Zf ( n ) in this differencerealization are the rational Hahn functions themselves.We can also recover contiguity relations for the rational functions U m ( n ; α, β, N ) using thisframework. We have γ ∗ n ( α ) γ ∗ ( α + 1) = αα − n (6.37)and hence from the last equality in (6.13) we see that X T d ∗ n ( x, α ) = − αd ∗ n ( x ; α + 1) . (6.38)21he relation XU m ( n ; α ) = ( e m ( x ; β ) , X T d ∗ n ( x ; α )) = αα − n ( e m ( x, α ) , d ∗ n ( x, α + 1)) = αα − n U m ( n ; α + 1)(6.39)gives the contiguity relation( n − α ) U m ( n ; α, β, N ) − n U m ( n − α, β, N ) = − α U m ( n ; α + 1 , β, N ) , (6.40)using the fact that U m ( n ) = ǫ m U m ( n ; α, β, N ) when the coefficients γ ∗ n given in (6.26) aresubstituted in (6.25). With the explicit expression (6.25) of U m ( n ; α, β, N ) as a F (1), wemay use the contiguity relation X U m ( n ; α, β, N ) = − α U m ( n ; α, β, N ) and its companion Z U m ( n ; α, β, N ) = αn − α U m ( n ; α, β, N ) that follows from (6.34), to obtain the tridiagonal rep-resentation given in (4.37), (4.38), (4.39), (4.40), (4.41), (4.42) of the algebra whose generators Z and X satisfy [ Z, X ] = Z + Z . It is also seen that Y U m ( n ; α ) =( e m ( x ; β ) , V T X T d ∗ ( x ; α )) = ( V e m ( x ; β ) , X T d ∗ ( x ; α )= − αν m ( e m ( x ; β ) , d ∗ ( x ; α + 1) = αn ( n − N + β ) U m ( n ; α + 1) , (6.41)which using the expression (6.35) for Y as a difference operator yields another contiguityrelation for U m ( n ; α, β, N ) (see [3]). The biorthogonal partners to the functions U m ( n ) are the scalar products ˜ U m ( n ) = ( e ∗ m , Zd n ).Using again the differential model we have˜ U m ( n ) = 12 πi I Γ dx γ n δ ∗ m x − n − N (1 − x ) − α F (cid:18) m − N, − m − β − N ; x (cid:19) . (6.42)Evaluating this integral amounts to identifying the coefficient of x N − n in the expansion(1 − x ) − α F (cid:18) m − N, − m − β − N ; x (cid:19) = 1 δ ∗ m X ℓ γ ℓ ˜ U m ( ℓ ) x N − ℓ (6.43)where again the domain of ˜ U m has been formally extended in the right-hand side for the sake ofnotation. To achieve this task, we may rely on the following fact which is derived in AppendixB. In view of the lower parameter − N , the polynomial F (cid:16) m − N, − m − β − N ; x (cid:17) is only definedfor powers of x up to N . This however suffices to identify the coefficient of x N − n in the lefthand side of (6.43). It so happens that although the functions (1 − x ) β F (cid:16) − m,m − N + β − N ; x (cid:17) and F (cid:16) m − N, − m − β − N ; x (cid:17) are manifestly not equal (one being a polynomial the other not), theycoincide term by term up to x N . Since this is the only range that matters for our computationwe might hence replace one by the other and look alternatively for the coefficient of x N − n in(1 − x ) − α + β F (cid:18) − m, m − N + β − N ; x (cid:19) = 1 δ ∗ m X ℓ γ ℓ ˜ U m ( ℓ ) x N − ℓ . (6.44)This turn out to be simpler since it has already been performed. Indeed comparing with (6.23),we see that δ ∗ m γ n ˜ U m ( n ) will be obtained from the expression for δ m γ ∗ n U m ( n ) given in (6.25) bymaking the substitutions n → ( N − n ) , α → ( β + 2 − α ) (see [3]). We thus find˜ U m ( n ) = δ ∗ m γ n ( α − β − N − n ( N − n )! F (cid:18) N − n, − m, m − N + β − N, β + 2 − α − n ; 1 (cid:19) . (6.45)22p to a normalization factor, this gives the biorthogonal partner [21] V m ( n ; α, β, N ) = U m ( N − n ; β + 2 − α, β, N ) (6.46)of the rational Hahn function U m ( n ; α, β, N ) multiplied by δ ∗ m γ n ( α − β − N − n ( N − n )! .We can now identify the weight function with respect to which these rational functions ofHahn type are orthogonal to one another. We have already established in Subsection 5.1.1 thebiorthogonality of the functions U m ( n ) and ˜ U k ( n ). Recall (6.18). Omitting the factor δ m δ ∗ k andother normalization constants, the biorthogonality relation P Nn =0 ˜ U k ( n ) U m ( n ) w n = 0 if k = m translates into N X n =0 ( α − β − N − n ( N − n )! (1 − α ) n n ! V k ( n ; α, β, N ) U m ( n ; α, β, N ) = 0 if k = m (6.47)using (6.25), (6.45), (1.4), (6.46) and (6.18). Note that the factor γ n γ ∗ n is cancelled throughthe product with w n . Using the identities (6.28), the first terms in (6.47) can be transformedto make this equation read N X n =0 V k ( n ; α, β, N ) U m ( n ; α, β, N ) w ( α,β ) n = 0 if k = m, (6.48)with w ( α,β ) n = ( β − α − N + 2) N ( β − N + 1) N ( − N ) n (1 − α ) n n !( β − α − N + 2) n . (6.49)We thus recover the weight function given in [3] which intervenes in the orthogonality of therational functions of Hahn type. The n -independent factors in (6.49) have been introduced sothat P Nn =0 w ( α,β ) n = 1. A central point we are making in this study is that the meta-Hahn algebra offers a unifieddescription of both the rational Hahn functions and the Hahn polynomials. In this sectionwe are using the differential model to confirm this and obtain explicitly the special functions.Having dealt with the rational functions, we now come to the orthogonal polynomials. Thesehave been identified in Subsection 5.2 with the scalar products S m ( n ) = ( e m , f ∗ n ) and ˜ S m ( n ) =( e ∗ m , f n ) where f n and f ∗ n are respectively the eigenvectors of the linear pencil W = X + µZ and its adjoint W T appropriately defined so that their common spectrum is (see Remark 4.1) ρ n = − n − α − µ. (6.50)Using the differential realization for X and Z given in (6.2) and (6.1) and the expressions (6.9)and (6.8) for their Lagrange adjoints, we find the following solutions for the EVPs W f n ( x ; µ ) = ρ n f n ( x ; µ ) and W T f ∗ n ( x ; µ ) = ρ n f ∗ n ( x ; µ ): f n ( x ; µ ) = ǫ n ( x − µ (cid:18) xx − (cid:19) n , (6.51) f ∗ n ( x ; µ ) = − ǫ ∗ n ( x − − µ − (cid:18) xx − (cid:19) − n − , n = 0 , . . . , N, (6.52)23here ǫ n and ǫ ∗ n are normalization constants. One readily verifies that the functions f ∗ k ( x ; µ )and f n ( x ; µ ) are mutually orthogonal:( f ∗ k ( x ; µ ) , f n ( x ; µ )) = − ǫ ∗ k ǫ n πi I Γ (1 − x ) − (cid:18) xx − (cid:19) n − k − dx = ǫ ∗ k ǫ n πi I Γ ′ y n − k − dy = ǫ ∗ n ǫ n δ k,n (6.53)using the change of variable y = xx − . (6.54)With e m ( x ; β ) and e ∗ m ( x ; β ) given by (6.19) and (6.20), we may now proceed to determine S m ( n ) and ˜ S m ( n ) as we have done for U m ( n ) and ˜ U m ( n ). Consider first S m ( n ) = ( f ∗ n ( x ; µ ) , e m ( x ; β )) (6.55)= − ǫ ∗ n δ m πi I Γ dx ( x − − µ − (cid:18) xx − (cid:19) − n − F (cid:18) − m, m − N + β − N ; x (cid:19) . In order to perform this integral it is useful to call upon the identity F (cid:18) − m, m − N + β − N ; x (cid:19) = ( − m ( x − m F (cid:18) − m, − m − β − N ; xx − (cid:19) (6.56)and to use the change of variable (6.54) to transform the integral in (6.55) into S m ( n ) = ( − m ǫ ∗ n δ m πi I Γ dy ( y − µ − m y − n − F (cid:18) − m, − m − β − N ; y (cid:19) . (6.57)In a familiar fashion, we see that the function S m ( n ) will be provided by the coefficient of y n in the expansion ( y − µ − m F (cid:18) − m, − m − β − N ; y (cid:19) = ( − m δ m X ℓ ǫ ∗ ℓ S m ( ℓ ) y ℓ (6.58)where the domain of S m is extended on the right hand side to avoid additional notation. Usingthe binomial theorem (6.24) and the explicit formula for the Gauss hypergeometric function,we find S m ( n ) = ( − m δ m ǫ ∗ n ( m − µ ) n n ! F (cid:18) − m, − m − β, − n − N, µ + 1 − m − n ; 1 (cid:19) . (6.59)At this point we bring the following transformation formula ∗ for the terminating F (1): F (cid:18) − n, b, cd, e ; 1 (cid:19) = ( e − c ) n ( e ) n F (cid:18) − n, d − b, cd, c − e − n + 1 ; 1 (cid:19) . (6.60)With the identification b = m + β − N, c = − m, d = − N, e = − µ, (6.61) ∗ This formula can be obtained from an example in [32] or from the Whipple formula [33] when one upperand one lower parameter in the terminating F (1) are set equal and sent to infinity.
24e have S m ( n ) = ( − m δ m ǫ ∗ n ( − µ ) n n ! F (cid:18) − m, m + β − N, − n − N, − µ ; 1 (cid:19) (6.62)and redefining the parameters µ = − ˆ α − , β = ˆ α + ˆ β + N + 1 (6.63)as in (4.48) and (4.49), we arrive at S m ( n ) = ( − m δ m ǫ ∗ n ( ˆ α + 1) n n ! Q m ( n, ˆ α, ˆ β, N ) (6.64)where Q m ( n, ˆ α, ˆ β, N ) are Hahn polynomials as per (1.1).In order to complete the picture, we shall examine the other related function: ˜ S m ( n ) =( e ∗ m , f n ). From (6.20) and (6.52) we have˜ S m ( n ) = ( f n ( x ; µ ) , e ∗ m ( x ; β )) (6.65)= ǫ n δ ∗ m πi I Γ dx ( x − µ − − N (cid:18) xx − (cid:19) − n − N F (cid:18) m − N, − m − β − N ; x (cid:19) . Performing the change of variable y = xx − one has˜ S m ( n ) = ǫ n δ ∗ m πi I Γ dy ( y − N − µ − y − n − N F (cid:18) m − N, − m − β − N ; yy − (cid:19) . (6.66)To compute this integral we need the coefficient of y N − n in the expansion of the two otherfactors of the integrand. This implies that the monomials in y of degree higher than N in thehypergeometric series will not affect the result. We may then use the fact, also explained inAppendix B, that F (cid:16) m − N, − m − β − N ; yy − (cid:17) coincides with (1 − y ) − m − β F (cid:16) − m, − m − β − N ; y (cid:17) up tothe power y N to replace one function by the other and look for the coefficient of y N − n in( y − N − µ − m − β − F (cid:18) − m, − m − β − N ; y (cid:19) = ( − m + β +1 δ ∗ m X ℓ ǫ N − ℓ S m ( N − ℓ ) y ℓ . (6.67)As it happens, we have carried out such an evaluation in (6.58). Upon comparing we see thatwe only need to do the substitutions: µ → N − µ − β − n → N − n (6.68)in the expression for ( − m δ m ǫ ∗ n S m ( n ) given in (6.62) to obtain the desired coefficient. This gives˜ S m ( n ) = ( − m + β +1 δ ∗ m ǫ n ( µ + β + 1 − N ) N − n ( N − n )! F (cid:18) − m, m + β − N, n − N − N, µ + β + 1 − N ; 1 (cid:19) . (6.69)Using the relations (6.63) we arrive at˜ S m ( n ) = ( − m + β +1 δ ∗ m ǫ n ( ˆ β + 1) N − n ( N − n )! F − m, m + ˆ α + ˆ β + 1 , n − N − N, ˆ β + 1 ; 1 ! . (6.70)25t can be seen from the difference equation obeyed by the Hahn polynomials that they havethe following property under the exchange n → N − n : Q m ( n, ˆ α, ˆ β, N ) = ( − m Q m ( N − n, ˆ β, ˆ α, N ) . (6.71)In light of this, we see that˜ S m ( n ) = ˆ δ ∗ m ǫ n ( ˆ β + 1) N − n ( N − n )! Q m ( n, ˆ α, ˆ β, N ) , (6.72)where signs have been absorbed in ˆ δ ∗ m . We thus verify what we have previously established ongeneral grounds: that S m ( n ) and ˜ S m ( n ) involve the same Hahn polynomials.As a by-product, we can readily obtain the weight function for the Hahn polynomials. Weknow from Subsection 5.2 that P Nn =0 ˜ S m ( n ) S k ( n ) ζ n = 0 if m = k , where ( f ∗ m , f n ) = ζ − n δ m,n .Here ζ − n = ǫ ∗ n ǫ n . Using (6.64) and (6.72) and dropping normalization factors, we have N X n =0 W (ˆ α, ˆ β ) n Q m ( n, ˆ α, ˆ β, N ) Q k ( n, ˆ α, ˆ β, N ) = 0 if m = k, (6.73)where W (ˆ α, ˆ β ) n = ( ˆ α + 1) n n ! ( ˆ β + 1) N − n ( N − n )! . (6.74)This expression coincides [4] (up to n -independent factors) with the standard weight functionassociated to the Hahn polynomials.Note that the differential model could be further exploited to obtain additional propertiesof the special functions connected to the meta-Hahn algebra. It will not have escaped thereader’s attention that this model has led to interesting representations of the Hahn rationalfunctions and Hahn polynomials in terms of contour integrals. m H algebra into the universalenvelopping algebra U ( sl ) We shall here make the observation that m H admits an embedding into U ( sl ). The Lie algebra sl has three generators { J , J ± } that satisfy the commutation relations[ J , J ± ] = ± J ± , [ J + , J − ] = 2 J . (7.1)Its Casimir element C ∈ U ( sl ) is given by C = J − J + J + J − . (7.2)In irreducible representations C is a multiple of the identity written in the form C = j ( j + 1) I and in finite-dimensional cases j = N where N + 1 is the dimension.We shall provide below the map of the meta-Hahn algebra into (the completion of) U ( sl ).To offer first the results of the computation in general, we shall consider the relations[ Z, X ] = Z + Z, (7.3)[ X, V ] = { V, Z } + η X + V + η Z + η I , (7.4)[ V, Z ] = η X + η Z + η I , (7.5)obtained from (2.1), (2.2), (2.3) after the constraints (2.5) imposed by the Jacobi identity havebeen implemented. 26 roposition 7.1 The following formulas provide an embedding of the algebra defined by (7.3) , (7.4) , (7.5) into U ( sl ) : Z = J + − I , X = − J J + + J + ξ J + + ξ I ,V = ξ J + ξ J + ξ J − + ξ I (7.6) with ξ = η − η η , ξ = − η , ξ = η + η ( ξ − / ,ξ = − η , ξ = 12 (cid:0) η ( ξ − ξ ) + η (1 − ξ ) − η (cid:1) (7.7) and the parameter ξ a root of the quadratic equation − η ξ + η ( − η + η ) ξ − η + 2 η η − η η + η η + η η + η C = 0 . (7.8)The Casimir operator (7.2) of the (non-standardized) meta-Hahn algebra takes the followingexpression under this embedding: Q = η C − η + η − ( η − η ) η . (7.9)We now revert to the standardized meta-Hahn algebra where η = 2 and η = − η (7.10)and moreover set η = 2 α − β − η and η to be η = ( N − − β ) α + β + 1, η = N − − β , as in (4.34) in the ( N + 1)-dimensional representationwhere C = N ( N + 1). It is then found that the coefficients ξ i , i = 0 , . . . ,
5, take the simpleexpressions: ξ = N − α, ξ = 1 − N , ξ = − , ξ = − β, ξ = − , ξ = N (cid:18) N − β (cid:19) . (7.11)and that the value of the Casimir element (2.13) is Q = − α − β + 2 α β − α . (7.12)Let us remark to conclude this Section that the differential operators (again keeping thesame notation for the abstract generators and their realizations): J = x ddx + τJ + = x,J − = − (cid:18) x d dx + 2 τ ddx (cid:19) , (7.13)verify the commutation relations (7.1) of sl with τ a parameter. Take τ = − N C = N ( N + 1)), it is readily observed that for this choice of τ , the differentialrealization (6.1), (6.2), (6.3) of m H at the heart of Section 6 is recovered when the differentialoperators (7.13) are substituted in the formulas (7.6) with the coefficients given by (7.11).27 Conclusion
We want to stress in closing that the analysis we have presented sets the stage for a novelalgebraic description of the polynomials of the Askey scheme coupled to associated biorthogonalrational functions. Before we comment on that let us first summarize the findings we reported.We introduced an algebra m H that we called the meta-Hahn algebra. The reason forthe name is that it admits embeddings of both the Hahn algebra and the rational Hahnalgebra which account for the bispectral properties of the Hahn orthogonal polynomials andrational functions respectively. The algebra m H thus unifies the algebraic description of thesetwo sets of functions. The algebra m H has three generators Z, X, V subjected to relationsthat involve at most one quadratic term. Its construction is centered around the subalgebragenerated by Z and X which is a real form of the deformed Jordan plane. With an eyeto the connection to special functions, certain elements of the representation theory of m H were developed. To that end, various bases of a finite-dimensional module were considered,namely the one associated to the GEVP defined by Z and X , the eigenbases of V and of thelinear pencil X + µZ and the corresponding adjoint bases. It was shown in particular that Z and X act tridiagonally in the eigenbasis of V and that V is a Hessenberg matrix in theGEVP basis. The features identified have allowed to establish that the overlaps between thesebases are biorthogonal rational functions or orthogonal polynomials that are bispectral. (Itwould be interest in the future to complete the study of the representation theory of m H .)A realization of m H in terms of differential operators was presented and used to obtain theexplicit expressions for the functions defined by these overlaps. The Hahn polynomials andrational functions were thus found and the model provided much of their characterization((bi-)orthogonality, contiguity relations, etc.) leading in particular to representations of thesefunctions in terms of contour integrals. It should be underscored moreover that the subalgebragenerated solely by Z and X is on its own intimately connected to the Hahn polynomials.Indeed its representations by tridiagonal matrices (which we have in the eigenbasis of V )were seen to entail the recurrence relations of these orthogonal polynomials. It was notedadditionally that m H derives from a potential and admits an embedding in U ( sl ). While theappendices that follow contain technical details, we want to point out that the computationsin Section 6 led to the identification of a Pad´e approximant for the binomial series in termsof Jacobi polynomials with a negative integer. This result which will be found in Appendix Bstems from traces of the Euler and Pfaff transformations of the hypergeometric series whichper se are not valid when the denominator parameter is a negative integer.Let us now explain why we believe that this study is paving the way for a new algebraic pic-ture of the Askey scheme extended in fact to include classes of biorthogonal rational functions.The meta-Hahn algebra can be viewed as an enlargement through the addition of the generator V of the non-commutative model of the plane given in this case by the algebra generated by Z and X subjected to [ Z, X ] = Z + Z . We have noted the connection of this algebra with theHahn polynomials. The commutation relations of V with X and Z are minimally quadraticand such that the resulting algebra admits an embedding of the Hahn algebra. Now, thereare a number of two-generated algebras and a classification can be found in [22], see also [34].Such algebras have arisen also in studies of martingale polynomials [35] and in the physicalcontext of asymmetric exclusion models [36]. We trust that these are related to hypergeomet-ric polynomials like our real version of the deformed Jordan plane. It is hence our convictionthat algebras analogous to the meta-Hahn algebras can be defined as extensions of these othertwo-dimensional algebras and that they will henceforth provide descriptions of families of or-thogonal polynomials and rational functions in a unified way. Let us mention the following in28upport of this. The q -oscillator algebra with generators X and Z verifying XZ − qZX = 1 isobviously one of the relevant two-generated algebras. Its tridiagonal representations have beenobtained in [21] and found to yield the recurrence coefficients of the Askey–Wilson polynomi-als. The analog of the generator V was also found and the resulting algebra was identified asthat of the big q -Jacobi polynomials. The exploration of the associated rational function wasnot carried out and this is obviously on our to-do list now. Another two-generated algebraof interest is the one with relation [ Z, X ] = Z + X which is discussed in [22]. Preliminaryinvestigations are indicating that this algebra is related to the Racah polynomials. We havetherefore good reasons to believe that the construction of an algebra analogous to the meta-Hahn algebra and centered around this “Racah” algebra will yield a framework offering anintegrated picture for the Racah polynomials and rational functions of the F -type. We arepoised to pursue this program and hope to report on it in the near future. A Orthogonality of the functions e n ( x ; β ) and e ∗ m ( x ; β ) We give in this Appendix the details of the computations showing that the functions e n ( x ; β )and e ∗ m ( x ; β ) defined in (6.19) and (6.20) are mutually orthogonal. The scalar product on V N has been defined by integration over the contour Γ taken to be a circle | x | = a < f ( x ) , g ( x )) = 12 πi I Γ f ( x ) g ( x ) dx, f, g ∈ V N . (A.1)We have( e ∗ m ( x ; β ) ,e n ( x ; β )) = δ ∗ m δ n πi I Γ dx x − − N F (cid:18) m − N, − m − β − N ; x (cid:19) F (cid:18) − n, n − N + β − N ; x (cid:19) (A.2)and want to show that ( e ∗ m ( x ; β ) , e n ( x ; β )) = 0 if m = n . Clearly, this property is equivalentto the statement that in the power expansionΨ nm ( x ) = F (cid:18) − m, m + β − N − N ; x (cid:19) F (cid:18) n − N, − n − β − N ; x (cid:19) = m − n + N X s =0 A s ( n, m ) x s (A.3)the coefficient of x N vanishes when n = m . We must therefore prove that A N ( n, m ) = 0 , n = m. (A.4)In order to show this, we consider product of the two terminating hypergeometric series in x :Ψ nm ( x ) = X s,i ( n − N ) s ( − n − β ) s ( − m ) i ( m + β − N ) i s ! i !( − N ) s ( − N ) i x s + i (A.5)to find that A N ( n, m ) = ( n − N ) N − i ( − n − β ) N − i ( − m ) i ( m + β − N ) i ( N − i )! i !( − N ) N − i ( − N ) i . (A.6)It is easily seen that the following product ( n − N ) N − i ( − m ) i of Pochhammer symbols is zerofor all i = 0 , , . . . , N if n = m . When n = m there is one and exactly one value of i for whichthis product is nonzero, it is i = n and then( n − N ) N − i ( − n ) i = − n !( N − n )! δ ni (A.7)We have thus have proven that A N ( n, m ) = 0 if n = m and that A N ( n, n ) = 0.29 Restricted versions of Euler’s and Pfaff ’s transformationsand a Pad´e approximation table for the binomial function
This Appendix confirms the validity of the substitutions that were performed in Subsubsection6.2.3 and Subsection 6.3 when computing ˜ U m ( n ) and ˜ S m ( n ). It offers restricted versions ofEuler’s and Pfaff’s transformations of the (truncating) hypergeometric series when the lowerparameter is a negative integer. A Pad´e approximant of the binomial function is moreoverobtained as a by-product.The Euler transformation of the hypergeometric series F (cid:18) a, bc ; x (cid:19) = (1 − x ) c − a − b F (cid:18) c − a, c − bc ; x (cid:19) (B.1)is not valid if c = − N , with N a nonnegative integer. Indeed, in this case the hypergeometricseries has singular terms. It is nevertheless possible to derive a restricted version of thistransformation that has proved most useful in the context of the differential model for m H .We have: Proposition B.1
Assume that
N, n are positive integers such that n < N and that b is a realnon-integer parameter. Then the relation F (cid:18) − n, b − N ; x (cid:19) = (1 − x ) − N + n − b F (cid:18) n − N, − N − b − N ; x (cid:19) (B.2) holds up to terms of degree N in x . The proof proceeds from multiplying the binomial series(1 − x ) β = ∞ X s =0 ( − β ) s s ! x s , β = − N + n − b (B.3)and the hypergeometric series F (cid:18) n − N, − N − b − N ; x (cid:19) = N − n X s =0 ( n − N ) s ( − N − b ) s s !( − N ) s x s . (B.4)Writing (1 − x ) − N + n − b F (cid:18) n − N, − N − b − N ; x (cid:19) = ∞ X k =0 A k x k , (B.5)the coefficients A k are found to be A k = ( N − n + b ) k k ! F (cid:18) − k, n − N, − N − b − N, − k − N + n − b ; x (cid:19) . (B.6)Clearly these A k are only defined for k ≤ N . The balanced hypergeometric F (1) seriesoccurring in (B.6) can then be evaluated using the Saalsch¨utz formula F (cid:18) − k, α, βγ, α + β + γ − k ; x (cid:19) = ( γ − α ) k ( γ − β ) k ( γ ) k ( γ − α − β ) k (B.7)30nd one thus establishes the validity of the proposition. Note that F (cid:16) − n,b − N ; x (cid:17) on theleft hand side of (B.2) is a polynomial in x of degree n while on the right hand side (1 − x ) − N + n − b F (cid:16) n − N, − N − b − N ; x (cid:17) is the product of a polynomial of degree N − n with the binomial(transcendental) function (1 − x ) − N + n − b and is therefore an infinite series in x . Nevertheless,truncating this series after the first N + 1 gives the polynomial A + A x + A x + . . . , + A N x N that coincides with the one on the left hand side. This is the precise meaning of PropositionB.1.A similar result that was used to compute the functions ˜ S m ( n ) arises when consideringPfaff’s transformation in the same context. We can formulate: Proposition B.2
Assume that
N, n are positive integers such that n < N and that b is a realnon-integer parameter. The relation (1 − x ) b F (cid:18) − n, b − N ; x (cid:19) = F (cid:18) n − N, b − N ; xx − (cid:19) (B.8) applies term by term as an identity between power series of x until the monomial x N is reached. This proposition can be considered as a restricted version of Pfaff’s transformation formula:(1 − x ) b F (cid:18) a, bc ; x (cid:19) = F (cid:18) c − a, bc ; xx − (cid:19) (B.9)which holds identically if c = − N . The proof is parallel to that of Proposition B.1. Oneexpands the left and right hand sides of (B.8) in power of x and uses the Saalsch¨utz formula(B.7) to observe the equality of both sides as long as the (resummed) coefficients are defined.There is a striking application of Proposition B.1 to the Pad´e approximation table of thebinomial function. It can be formulated as follows: Proposition B.3
Let β be an arbitrary non-integer parameter. The first N + 1 terms of thebinomial series (1 − x ) β = ∞ X s =0 ( − β ) s s ! x s (B.10) coincide with the first N + 1 terms of the power expansion of the rational function R ( x ) = F (cid:16) n − N, − n − β − N ; x (cid:17) F (cid:16) − n,n + β − N − N ; x (cid:17) = N X s =0 ( − β ) s s ! x s + O ( x N +1 ) . (B.11)This proposition is an immediate corollary of Proposition B.1. It allows to construct the wholePad´e interpolation table (see for example [37]) for the binomial function. Namely, we havethat for every non-negative integers n, m , the binomial series coincides with the first n + m + 1terms of the power expansion of the rational function R mn ( x ) = F (cid:16) − m, − n − β − n − m ; x (cid:17) F (cid:16) − n, − m + β − n − m ; x (cid:17) . (B.12)Hence the set of the rational functions R mn ( x ) for all m, n = 0 , , , . . . constitutes the Pad´einterpolation table for the binomial function (1 − x ) β .31 cknowledgments The authors express their thanks to Andr´e Beaudoin, Geoffroy Bergeron, Antoine Brillant,Maxim Derevyagin, Julien Gaboriaud and Satoshi Tsujimoto for enlightening discussions.The work of LV is supported in part by a Discovery Grant from the Natural Sciences andEngineering Research Council (NSERC) of Canada. AZ is gratefully holding Simons CRMProfessorship and is funded by the National Foundation of China (Grant No.11771015).
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