A set of q-coherent states for the Rogers-Szegö oscillator
aa r X i v : . [ m a t h - ph ] F e b A set of q -coherent states for the Rogers-Szeg¨o oscillator Othmane El Moize ♭ , Zouha¨ır Mouayn ∗ ∗ Department of Mathematics, Faculty of Sciences,Ibn Tofa¨ıl University, P.O. Box. 133, K´enitra, Morocco ♭ Department of Mathematics, Faculty of Sciences and Technics (M’Ghila),Sultan Moulay Slimane University, P.O. Box. 523, B´eni Mellal, Morocco
Abstract
We discuss a model of a q -harmonic oscillator based on Rogers-Szeg¨o functions. We combine these functions with aclass of q -analogs of complex Hermite polynomials to construct a new set of coherent states depending on a nonnegativeinteger parameter m . Our construction leads to a new q -deformation of the m -true-polyanalytic Bargmann transformwhose range defines a generalization of the Arik-Coon space. We also give an explicit formula for the reproducing kernelof this space. The obtained results may be exploited to define a q -deformation of the Ginibre- m -type process on thecomplex plane. February 23, 2021
Coherent states (CS) are a type of quantum states which were first introduced by Schrodinger [1] whenhe described certain states of the harmonic oscillator (HO). Later, Glauber [2] used these states for hisquantum mechanical description of coherent laser light, and coined the term CS. Like canonical CS, theysatisfy a set of relevant properties and they too have found wide applications in different branches ofphysics such as quantum optics, statistical mechanics, nuclear physics and condensed matter physics [3].One of many definitions of CS is a special superposition with the formΨ z := (cid:0) e z ¯ z (cid:1) − / ∞ X j =0 ¯ z j √ j ! φ j , (1.1)where the φ j ’s span a Hilbert space H usually called Fock space.The terminology of generalized CS (GCS) was first appeared and studied in [4, 5] in connectionwith states discussed in [6]. These states are usually associated with potential algebras other than theoscillator one [3, 7, 8]. An important example is provided by the q -deformed CS ( q -CS for brevity) relatedto deformations of boson operators [9, 10, 11]. q -CS are usually constructed in a way that they reduce totheir standard counterparts as q → f -deformedCS [12]. Among q -CS, there are those associated with the relation a q a ∗ q − qa ∗ q a q = 1 (1.2)with 0 < q <
1, where a q are often termed maths-type q -bosons operators [13, 14] because the basic numbers and special functions attached to them have been extensively studied in mathematics [15]. Asa consequence, the corresponding wavefunctions of these q -boson operators were found in terms of manyorthogonal q -polynomials [16, 17, 18]. 1he q -CS may be defined [9] through a q -analog of the number states expansion (1.1) asΨ qz := ( e q ( z ¯ z )) − + ∞ X j =0 ¯ z j p [ j ] q ! φ ( q ) j , (1.3)for z ∈ C q := { ζ ∈ C , (1 − q ) ζ ¯ ζ < } where e q ( z ¯ z ) being a q -exponential function (see Eq. (2.7) below).Here, the φ ( q ) j ’s span a Hilbert space H q which stands for a q -analog of the Fock space H .In this paper, we adopt the Hilbertian probabilistic formalism [19] in order to generalize the expressionin (1.3) with respect to a fixed integer parameter m ∈ N . Precisely, we here introduce the followingsuperposition of the φ ( q ) j ’s: Ψ q,mz := ( N q,m ( z ¯ z )) − + ∞ X j =0 Φ q,mj ( z ) φ ( q ) j , (1.4)where N q,m ( z ¯ z ) is a normalization factor given by Eq. (6.2) below andΦ q,mj ( z ) := q ( m ∧ j ) p q m (1 − q ) | m − j | | z | | m − j | e − i ( m − j ) arg ( z ) (( q ; q ) m ∨ j ) − ( − m ∧ j ( q ; q ) | m − j | p q mj ( q ; q ) m ( q ; q ) j P m ∧ j (cid:16) q m (1 − q ) z ¯ z ; q | m − j | | q (cid:17) (1.5)are coefficients defined in terms of Wall polynomials P n ( · , a | q ) [20], where m ∨ j = max( m, j ) and m ∧ j =min( m, j ). In the case m = 0, Φ q, j ( z ) reduce to the coefficients z j / q [ j ] q !. For general m ∈ N , thecoefficients (1.5) are a slight modification of a class of 2 D orthogonal q -polynomials, denoted H j,m ( z, z | q ) , which are q -deformation [21] of the well known complex Hermite polynomials [22] H j,m ( z, z ) whoseRodriguez-type formula reads H j,m ( z, z ) = ( − j + m e zz ∂ jz ∂ mz e − zz . This last expression turns out to bethe diagonal representation (or upper symbol [23] with respect to CS (1.1)) of the operator ( a ∗ ) j a m whose expectation values are needed [24] in the study of squeezing properties involving the Heisenberguncertainty relation. Here, a and a ∗ are the classical annihilation and creation operators. In other words,the polynomial H j,m ( z, z ) also represents a classical observable on the phase space or a Glauber-Sudarshan P -function for ( a ∗ ) j a m . This means that the G q -CS Ψ q,mz in (1.4) we are introducing may play an analogrole in the Berezin de -quantization procedure of (cid:0) a ∗ q (cid:1) j a mq , see [25] for a similar discussion related the q -Heisenberg-Weyl group.Here, we introduce an explicit realization of q -creation and q -annihilation operators associated to aRogers-Szeg¨o oscillator whose eigenstates are chosen to be our vectors φ ( q ) j . Next, we give a closed formfor the superposition (1.4) and the associated CS transform (CST) B ( q ) m which may be viewed as a new q -deformation of the m -true polyanalytic Bargmann transform [26, 27]. As a consequence, the range of B ( q ) m defines a generalization of the well known Arik-Coon space [9], whose reproducing kernel will begiven explicitly. The latter one may be used to define a q -deformation of the Ginibre- m -type process onthe complex plane [28].The paper is organized as follows. In section 2, we recall some notations of q -calculus. In Section3, we give a brief review of the standard coherent state formalism. In section 4, we discuss the Hilbertspace carrying the coefficients needed in the superposition defining our G q -CS. Section 5 is devoted todiscuss a model of a q -deformed HO based on Rogers-Szeg¨o functions with a realization of q -creationand q -annihilation operators on L ( R ). In Section 6, we introduce a new set of G q -CS for the Rogers-Szeg¨o oscillator and we give an explicit formula for the associated CST whose range provides us with ageneralization of the Arik-Coon space. For the latter one we also obtain explicitly the reproducing kernel.Most technical proofs and calculations are postponed in Appendices.2 Notations
This section collects the basic notations of q -calculus and definitions used in the rest of the paper. Thereader may proceed to Section 3 and refer back here as necessary. For more details we refer to [15, 20, 30].We assume that 0 < q < For a ∈ C , we have the following definitions. The number[ a ] q = 1 − q a − q (2.1)is called a q -number and satisfies [ a ] q → a as q →
1. In particular, [ n ] q is known as q -integer. The q -shifted factorial is defined by( a ; q ) = 1 , ( a ; q ) n := n − Y k =0 (cid:16) − aq k (cid:17) , n ∈ N , ( a ; q ) ∞ := ∞ Y k =0 (cid:16) − aq k (cid:17) , (2.2)and, for any α ∈ C , we shall also use( a ; q ) α = ( a ; q ) ∞ ( aq α ; q ) ∞ , aq α = q − n , n ∈ N . (2.3)For a , a , ..., a m ∈ C , the multiple q -shifted factorials is defined as follows( a , a , ..., a l ; q ) n = ( a ; q ) n ( a ; q ) n · · · ( a l ; q ) n , l ∈ N , n ∈ N ∪ {∞} . (2.4) The q -binomial coefficient is given by (cid:20) nk (cid:21) q := [ n ] q ![ n − k ] q ![ k ] q ! = ( q ; q ) n ( q ; q ) n − k ( q ; q ) k , k = 0 , , · · · , n, (2.5)where [ n ] q ! = ( q ; q ) n (1 − q ) n (2.6)denotes the q -factorial of n ∈ N . A q -analog for the exponential function is given by e q ( z ) := X n ≥ z n [ n ] q ! = 1((1 − q ) z ; q ) ∞ , | z | < − q , (2.7) The basic hypergeometric series is defined by r φ s (cid:18) a , ..., a r b , ..., b s | q ; z (cid:19) := ∞ X k =0 ( a , ..., a r ; q ) k ( b , ..., b s ; q ) k ( − (1+ s − r ) k q (1+ s − r ) ( k ) z k ( q ; q ) k . (2.8)We note that the series r φ s converges absolutely for all z if r ≤ s and for | z | < r = s + 1. In thisspacial case Eq.(2.8) reduces to s +1 φ s (cid:18) a , ..., a s +1 b , ..., b s | q ; z (cid:19) := ∞ X k =0 ( a , ..., a s +1 ; q ) k ( b , ..., b s ; q ) k z k ( q ; q ) k . (2.9)3 A CS formalism
Here, we adopt the prototypical model of CS presented in ([19], pp.72-77) and described as follows. Let X be a set equipped with a measure dν and L ( X, dµ ) the Hilbert space of dµ -square integrable functions f ( x ) on X . Let A ⊂ L ( X, dµ ) be a subspace with an orthonormal basis { Φ k } ∞ k =0 such that N ( x ) := X j ≥ | Φ j ( x ) | < + ∞ , x ∈ X. (3.1)Let H be another (functional) Hilbert space with the same dimension as A and { φ k } ∞ k =0 is a givenorthonormal basis of H . Then, consider the family of states {| x i} x ∈ X in H , through the following linearsuperpositions | x i := ( N ( x )) − / X j ≥ Φ j ( x ) ϕ j . (3.2)These CS obey the normalization condition h x | x i H = 1 (3.3)and the following resolution of the unity in H H = Z X | x ih x |N ( x ) dµ ( x ) (3.4)which is expressed in terms of Dirac’s bra-ket notation | x ih x | meaning the rank one operator defined by ϕ
7→ h x | ϕ i H | x i , ϕ ∈ H . The choice of the Hilbert space H defines in fact a quantization of the space X by the coherent states in (3.2), via the inclusion map X ∋ x
7→ | x i ∈ H and the property (3.4) is crucialin setting the bridge between the classical and the quantum worlds. The CS transform (CST) associatedwith the set | x i is the map B : H −→ A defined for every x ∈ X by B [ φ ]( x ) := ( N ( x )) / h φ | x i H . (3.5)From the resolution of the identity (3.4) and for φ, ψ ∈ H , we have h φ, ψ i H = hB [ φ ] , B [ ψ ] i L ( X ) (3.6)meaning that B is an isometric map.The formula (3.2) can be considered as a generalization of the series expansion of the canonical CS | z i = (cid:0) e z ¯ z (cid:1) − / X j ≥ z j √ j ! φ j , z ∈ C (3.7)with { φ j } ∞ j =0 being an orthonormal basis of the Hilbert space H := L ( R ), consisting of eigenstates of thequantum harmonic oscillator given by φ j ( ξ ) = (cid:0) √ π j j ! (cid:1) − / H j ( ξ ) e − ξ (3.8)where H j ( ξ ) := j ! ⌊ j/ ⌋ X k =0 ( − k k !( j − k )! (2 ξ ) j − k , ξ ∈ R (3.9)is the Hermite polynomial of degree j ([20], p.59). Here, the space A is the Fock space F ( C ) of entirecomplex-valued functions which are π − e − z ¯ z dλ -square integrable and N ( z ) = e z ¯ z , z ∈ C . Here dλ denotesthe Lebesgue measure on C ∼ = R . In this case, the associated CST B : L ( R ) → F ( C ), defined for anyfunction f ∈ L ( R ) by B [ f ]( z ) := π − Z R e − ξ + √ ξz − z f ( ξ ) dξ, z ∈ C (3.10)turns out to be the well known Bargmann transform ([29], p.12).4 The Hilbert space F q We observe that the coefficients Φ qj ( z ) := z j p [ j ] q ! , j = 0 , , , . . . , (4.1)occurring in the number state expansion of Ψ qz in Eq. (1.3) constitute a particular case of the complex-valued functions :Φ q,mj ( z ) := q ( m ∧ j ) √ − q | m − j | | z | | m − j | e − i ( m − j )arg(z) (( q ; q ) m ∨ j ) − ( − m ∧ j ( q ; q ) | m − j | p q mj ( q ; q ) m ( q ; q ) j P m ∧ j (cid:16) (1 − q ) z ¯ z ; q | m − j | | q (cid:17) (4.2)where P n ( x ; q s | q ) denotes the Wall polynomial defined by ([20], p.109) P n ( x ; a | q ) = φ (cid:18) q − n , aq | q ; qx (cid:19) = 1( a − q − n ; q ) n φ (cid:18) q − n , x − − (cid:12)(cid:12)(cid:12) q ; xa (cid:19) . (4.3)Indeed, for m = 0 we have that Φ q, j ( z ) = Φ qj ( z ). For m ∈ N , these coefficients are a slight modificationof the 2 D complex orthogonal polynomials ([21], p.4) : H m,j ( z, ζ | q ) := m ∧ j X k =0 (cid:20) mk (cid:21) q (cid:20) jk (cid:21) q ( − k q ( k )( q ; q ) k z m − k ζ j − k , z, ζ ∈ C . (4.4)Precisely, Φ q,mj ( z ) = H m,j ( √ − qz, √ − q ¯ z | q ) p q mj ( q ; q ) j ( q ; q ) m . (4.5)Note, also, that by using the identity (2.6), one can check that lim q → Φ q,mj ( z ) = ( m ! j !) − / H m,j ( z, ¯ z ) where H r,s ( z, ζ ) = m ∧ j X k =0 ( − k k ! (cid:18) mk (cid:19)(cid:18) jk (cid:19) z m − k ζ j − k (4.6)denote the 2 D complex Hermite polynomials introduced by Itˆo [22] in the context of complex Markovprocesses.Furthermore, we observe that the functions (4.2) constitute an orthonormal system in the Hilbertspace F q := L ( C q,m , dµ q ( z )) of square integrable functions on C q,m := { ζ ∈ C , (1 − q ) ζ ¯ ζ < q m } withrespect to the measure dµ q ( z ) = dθ π ⊗ X l ≥ q l ( q ; q ) ∞ ( q ; q ) l δ ( r − q l √ − q ) , (4.7) z = re iθ , r ∈ R + , θ ∈ [0 , π ) and δ ( r − • ) is the Dirac mass function at the point • . Indeed, we mayassume that m ≥ j because of the symmetry H r,s ( z, w | q ) = H s,r ( w, z | q ), then we use the expression (4.2),we obtain h Φ q,mj , Φ q,mk i F q = Z C q,m Φ q,mj ( z )Φ q,mk ( z ) dµ q ( z )= ( − j + k ( q ; q ) m q ( j ) √ − q m − j ( q ; q ) m − j ( q ; q ) m p q mj ( q ; q ) j ( q ; q ) m q ( k ) √ − q m − k ( q ; q ) m − k p q mk ( q ; q ) k Z π e iθ ( j + k − m ) dθ π × ∞ X l =0 q l ( q ; q ) ∞ ( q ; q ) l P j ( r , q m − j | q ) P k ( r , q m − k | q ) r m − j − k δ ( r − q l √ − q )5 ( − j + k ( q ; q ) m q ( j ) + ( k ) − mj ( q ; q ) j ( q ; q ) m − j ( q ; q ) m − k δ j,k ∞ X l =0 q l (1+ m − j ) ( q ; q ) ∞ ( q ; q ) l P j ( q l , q m − j | q ) P k ( q l , q m − j | q ) = δ j,k . Here, we have used the orthogonality relations of Wall polynomials ([20], p.107): ∞ X l =0 ( τ q ) l ( q ; q ) l P s ( q l ; τ | q ) P n ( q l ; τ | q ) = ( τ q ) n ( τ q ; q ) ∞ ( q ; q ) n ( τ q ; q ) n δ s,n , < τ < q − (4.8)for parameters τ = q m − j , s = j and n = k .Finally, we denote by A q ( C ) the completed space of holomorphic functions on C q , equipped with thescalar product h ϕ, φ i = Z C q ϕ ( z ) φ ( z ) dµ q ( z ) . (4.9)The element Φ q, j ( z ) = ([ j ] q !) − / z j form an orthonormal basis of the space A q ( C ) which co¨ıncides withArik-Coon space [9] whose reproducing kernel is given by K q ( z, w ) := e q ( z ¯ w ) , z, w ∈ C q . (4.10)Moreover, by letting q →
1, the measure dµ q reduces to the Gaussian measure π − e − z ¯ z dλ on C . Following [31], we consider a q -deformed creation and annihilation operators as follows: B ∗ q = e iκx i √ − q (cid:16) e iκx − q / e iκ∂ x (cid:17) (5.1) B q = − e − iκx i √ − q (cid:16) e − iκx − q / e iκ∂ x (cid:17) . (5.2)Here κ is a deformation parameter related to a finite-difference method with respect to x and q = e − κ , κ >
0. Note that for a ∈ C the operator e a∂ x acts on a function f ( x ) as e a∂ x [ f ]( x ) = f ( x + a ) . Moreover, it is not difficult to verify that the operators (5.1)-(5.2) satisfy the q -commutation relation[ B q , B ∗ q ] = B q B ∗ q − qB ∗ q B q = 1 . (5.3)Similarly to the case of the quantum linear harmonic oscillator, the q -deformed harmonic oscillator isdescribed by the Hamiltonian H q = 12 ~ ω (cid:0) B q B ∗ q + B ∗ q B q (cid:1) (5.4)where ω is the oscillator frequency and ~ denotes the Planck’s constant. Explicitly, H q = 12( q − ~ ω (cid:16) − q / + q / ) e iκx e iκ∂ x + ( q − / + q / ) e − iκx e iκ∂ x + ( q / + q / ) e iκ∂ x (cid:17) . (5.5)For the sake of simplicity we will take ~ = ω = 1. The eigenstates of the the Hamiltonian in (5.5) aregiven by ϕ RSj ( x ) := ( i √ q ) j π p ( q ; q ) j H j (cid:16) − q − e iκx ; q (cid:17) e − x , x ∈ R (5.6)in terms of the Rogers–Szeg¨o polynomials [32] H n ( ξ ; q ) := n X k =0 (cid:20) nk (cid:21) q ξ k . (5.7)6oreover, in [31] the authors showed that the functions (5.6) satisfy an orthonormality relation on thefull real line, i.e. Z R ϕ RSj ( x ) ϕ RSk ( x ) dx = δ jk . (5.8)Furthermore, each function ϕ RSj can also be reproduced by iterating j times the action of the q -creationoperator B † q on the Gaussian function as ϕ RSj ( x ) = (cid:0) B ∗ q (cid:1) j (cid:18) π − e − x (cid:19) . (5.9)So, one can check that the q -deformed creation and annihilation operators act on the q -wave functions ϕ RSj ( x ) as: B ∗ q ϕ RSj ( x ) = ϕ RSj +1 ( x ) and B q ϕ RSj ( x ) = [ j ] q ϕ RSj − ( x ) . (5.10)Then, it follows that the Hamiltonian (5.4) is diagonal on the states ϕ RSj ( x ) and has the eigenvalues ε qj = 12 ([ j + 1] q + [ j ] q ) (5.11)as q -deformed energy levels.Finally, let us mention that in the limit q → κ → q → ϕ RSj ( x ) = (cid:0) √ π j j ! (cid:1) − / H j ( x ) e − x (5.12)where H j ( · ) is the Hermite polynomial, meaning that they stand for a q -analogs of the linear harmonicoscillator wave functions which justify our choice in (5.6). q -CS For m ∈ N and q ∈ ]0 , q -CS by the following superposition of the ϕ RSj ’s:Ψ q,mz := ( N q,m ( z ¯ z )) − X j ≥ Φ q,mj ( z ) ϕ RSj (6.1)where N q,m ( z ¯ z ) = q − m ( q − m (1 − q ) z ¯ z ; q ) m ( q − m (1 − q ) z ¯ z ; q ) ∞ (6.2)is the normalization constant ensuring h Ψ q,mz | Ψ q,mz i L ( R ) = 1 and well defined for z ∈ C q,m .To check that states Ψ q,mz provide us with a resolution of the identity operator L ( R ) , we proceed bywriting the operator-valued integralˆ O q,m := Z C N q,m ( z ¯ z ) dµ q ( z ) | Ψ q,mz ih Ψ q,mz | , (6.3)where the Dirac’s bra-ket notation | Ψ ih Ψ | means the rank-one operator ϕ Ψ | ϕ i · Ψ, ϕ ∈ L ( R ).Replacing in Eq.(6.3) the state Ψ q,mz by its expression defined by Eq.(6.1) and using polar coordinates z = re iθ , r ∈ R + , θ ∈ [0 , π ), then the operator ˆ O q,m decomposes asˆ O q,m = ( q ; q ) ∞ ( q ; q ) m + ∞ X j,k =0 ( − m ∧ j + m ∧ k q ( m ∧ j ) + ( m ∧ k )( q ; q ) m ∨ j ( q ; q ) m ∨ k ( q ; q ) | m − j | ( q ; q ) | m − k | q q m ( j + k ) ( q ; q ) j ( q ; q ) k "(cid:18)Z π e i ( j − k ) θ dθ π (cid:19) ∞ X l =0 q l (1+ ( | m − j | + | m − k | )) ( q ; q ) l P m ∧ j ( q l ; q | m − j | | q ) P m ∧ k ( q l ; q | m − k | | q ) | ϕ RSj ih ϕ RSk | . (6.4)By using direct calculations, Eq.(6.4) can be written asˆ O q,m = ∞ X j =0 ( q ; q ) ∞ ( q ; q ) m q j ( j − − m ) ( q ; q ) m − j ( q ; q ) m − j ( q ; q ) j " ∞ X l =0 q l (1+ m − j ) ( q ; q ) l (cid:16) P j ( q l ; q m − j | q ) (cid:17) | ϕ RSj ih ϕ RSk | . (6.5)Finally, applying the orthogonality relations of Wall polynomials (4.8) for parameters τ = q m − j and s = n = j , we arrive, after some simplifications, atˆ O q,m = ∞ X j =0 | ϕ RSj ih ϕ RSj | = L ( R ) . (6.6)In Eq.(6.3) the integration is restricted to the part C q,m of the complex plane where the normalizationgiven by Eq.(6.2) converges.Now, we establish (see Appendix A for the proof) that the wave function of the CS (6.1) can beexpressed asΨ q,mz ( ξ ) = ( − m q m e − ξ ( q − m (1 − q ) z ¯ z ; q ) ∞ √ π ( q ; q ) m ( q − m (1 − q ) z ¯ z ; q ) m ! q − m/ e imκξ ( iz q − qq m − , − iz q − qq m e iκξ ; q ) ∞ × Q m ie − iκξ q / − ie iκξ q − / zq − / q − qq m − e iκξ , ¯ zq / q − qq m − e − iκξ | q ! , ξ ∈ R (6.7)in terms of the AL-Salam-Chihara polynomials Q m ( . ; ., . | q ), which are defined by ([20], p.80): Q m ( x ; α, β | q ) = ( αβ ; q ) m α m φ (cid:18) q − m , αu, αu − αβ, (cid:12)(cid:12)(cid:12) q ; q (cid:19) , x = 12 ( u + u − ) . (6.8)Further, by applying (3.5), the CST B qm : L ( R ) −→ L ( C q,m , dµ q ( z )) defined by B qm [ f ]( z ) = ( N q,m ( z ¯ z )) h f, Ψ q,mz i L ( R ) , z ∈ C q,m , (6.9)is an isometric map. Explicitly, it is given by B qm [ f ]( z ) = q − m/ √ π ( q ; q ) m ! ( − m ( iz q − qq m − ; q ) ∞ × Z R e imκξ e − ξ ( − iz q − qq m e iκξ ; q ) ∞ Q m ie − iκξ q / − ie iκξ q − / zq − / q − qq m − e iκξ , ¯ zq / q − qq m − e − iκξ | q ! f ( ξ ) dξ (6.10)for every z ∈ C q,m .Particularly, for m = 0, the wave functions of the CS in (6.7) has the formΨ q, z ( ξ ) = ( e q ( z ¯ z )) − π − ( − iz √ − qe iκξ ; q ) ∞ e − ξ ( iz p q (1 − q ); q ) ∞ , (6.11)8here ξ ∈ R and z ∈ C q := C q, which is the domain of convergence of the exponential function in (2.7).In this case, the transform (6.10) reduces to B q : L ( R ) −→ A q ( C ), defined by B q [ f ]( z ) = π − ( iz p q (1 − q ); q ) ∞ Z R e − ξ ( − iz √ − qe iκξ ; q ) ∞ f ( ξ ) dξ. (6.12)for every z ∈ C q . Moreover, when 1 ← q , B q goes to the Bargmann transform (3.10).In addition, by letting q → B qm goes to the generalized Bargmann transform B m : L ( R ) → A m ( C ) ⊂ L ( C , π − e − z ¯ z dλ ), defined by [26]: B m [ f ]( z ) = ( − m (2 m m ! √ π ) − Z R e − z − ξ + √ ξz H m (cid:18) ξ − z + ¯ z (cid:19) f ( ξ ) dξ (6.13)where H m ( . ) denotes the Hermite polynomial, see Appendix B for the proof. Here, A m ( C ) is the gener-alized Bargmann Fock space whose reproducing kernel is given by [33]: K m ( z, w ) = e z ¯ w L (0) m (cid:0) | z − w | (cid:1) , z, w ∈ C . (6.14)Consequently, the range of the CST (6.10) leads us to define a generalization , with respect to m ∈ N ,of the Arik-Coon space A q as A q,m = B qm ( L ( R )) which also coincides with the closure in F q of the linearspan of { Φ q,mj } j ≥ . In Appendix C, we prove that the reproducing kernel of the space A q,m is given by K q,m ( z, w ) = ( q − m (1 − q ) z ¯ z ; q ) m q m ( q − m (1 − q ) z ¯ w ; q ) ∞ φ q − m , zw , q ¯ z ¯ wq, q − m +1 (1 − q ) z ¯ z (cid:12)(cid:12)(cid:12) q ; q (1 − q ) w ¯ w ! (6.15)for every z, w ∈ C q,m . In particular, the limitlim q → K q,m ( z, w ) = K m ( z, w ) (6.16)holds true.Finally, note that the expression (6.15) may also constitute a starting point to construct a q -deformationfor the determinantal point process associated with an m th Euclidean Landau level or Ginibre-type pointprocess in C as discussed by Shirai [28]. Remark.
Note that the Stieltjes-Wigert polynomials are defined as ([20], p.116): s n ( x ; q ) := n X k =0 (cid:20) nk (cid:21) q q k x k (6.17)and are connected to the Rogers-Szeg¨o polynomials (5.7) by s n ( x ; q − ) = H n ( xq − n ; q ) . (6.18)This last relation may also suggest the construction of another extension of q -deformed CS via the sameprocedure. The constructed G q -CS would be related to a suitable Arik–Coon oscillator with q > ppendix A By using (6.1) and replacing Φ m,qj ( z ) by their expressions in (4.2), we need the closed form of the followingseries S m = X j ≥ ( − m ∧ j ( q ; q ) m ∨ j q ( m ∧ j ) √ − q | m − j | | z | | m − j | e − i ( m − j ) θ ( q ; q ) | m − j | p q mj ( q ; q ) m ( q ; q ) j P m ∧ j (cid:16) (1 − q ) z ¯ z ; q | m − j | | q (cid:17) ϕ RSj ( ξ ) . This sum can be cast into two quantities as S m ( < ∞ ) + S m ( ∞ ) where S m ( < ∞ ) = m − X j =0 ( − j ( q ; q ) m q ( j ) √ − q m − j ¯ z m − j ( q ; q ) m − j p q mj ( q ; q ) m ( q ; q ) j P j (cid:0) (1 − q ) z ¯ z ; q m − j | q (cid:1) ϕ RSj ( ξ ) − m − X j =0 ( − m ( q ; q ) j q ( m ) √ − q j − m z j − m ( q ; q ) j − m p q mj ( q ; q ) m ( q ; q ) j P m (cid:0) (1 − q ) z ¯ z ; q j − m | q (cid:1) ϕ RSj ( ξ ) , (6.19)and S m ( ∞ ) ( z, ξ ; q ) = ( − m q ( m ) ( z √ − q ) − m e − ξ π p ( q ; q ) m η ( ∞ ) ( m, z, ξ ) , (6.20)with η ( ∞ ) ( m, z, ξ ) = ∞ X j =0 ( z √ − q ) j p q mj ( q ; q ) j − m P m ((1 − q ) z ¯ z ; q j − m | q ) ( i √ q ) j H j (cid:16) − q − e iκξ ; q (cid:17) . (6.21)Now, we apply the relation ([35], p.3) : P n ( x ; q − N | q ) = x N ( − − N q N ( N +1 − n )2 ( q N +1 ; q ) n − N ( q − N ; q ) n P n − N ( x ; q N | q ) (6.22)for N = j − m, n = j and x = (1 − q ) z ¯ z , to obtain that S m ( < ∞ ) = 0 . For the infinite sum, let us rewritethe Wall polynomial as ([20], p.260) : P n ( x ; a | q ) = ( x − ; q ) n ( aq ; q ) n ( − x ) n q − ( n ) φ (cid:18) q − n , xq − n (cid:12)(cid:12) q ; aq n +1 (cid:19) , (6.23)and by using the identity ( b ; q ) n = ( b − q − n ; q ) n ( − b ) n q ( n ) , (6.24)for b = x − , the polynomial (6.23) reduces to P n ( x ; a | q ) = ( xq − n ; q ) n ( aq ; q ) n φ (cid:18) q − n , xq − n (cid:12)(cid:12) q ; aq n +1 (cid:19) . (6.25)Thus, for n = m, x = (1 − q ) z ¯ z and a = q j − m in (6.25), and by making appeal to the identity( a ; q ) n + k = ( a ; q ) n ( aq n ; q ) k , (6.26)the r.h.s of (6.21) takes the form η ( ∞ ) ( m, z, ξ ) = ( q − m λ ; q ) m ∞ X j =0 y j ( q ; q ) j H j ( α ; q ) φ (cid:18) q − m , λq − m (cid:12)(cid:12) q ; q j (cid:19) (6.27)10here y = iz r − qq m − , α = − q − e iκξ and λ = (1 − q ) z ¯ z . Using the definition of φ as in (2.9) andinterchanging the order of summation, we get that η ( ∞ ) ( m, z, ξ ) = ( q − m λ ; q ) m ∞ X k =0 ( q − m ; q ) k ( λq m − ; q ) k q k ( q ; q ) k ∞ X j =0 ( yq k ) j ( q ; q ) j H j ( α ; q ) . (6.28)By using the generating function of the Rogers-Szeg¨o polynomials ([36], p.1): X j ≥ H j ( x ; q )( q ; q ) j t j = 1( t ; q ) ∞ ( xt ; q ) ∞ (6.29)for t = yq k and x = α , (6.28) becomes η ( ∞ ) ( m, z, ξ ) = ( q − m λ ; q ) m ∞ X k =0 ( q − m ; q ) k ( λq m − ; q ) k q k ( q ; q ) k yq k ; q ) ∞ ( αyq k ; q ) ∞ . (6.30)By applying (2.3), it follows that η ( ∞ ) ( m, z, ξ ) = ( q − m λ ; q ) m ( y, αy ; q ) ∞ ∞ X k =0 ( q − m , αy, y ; q ) k ( λq m − ; q ) k q k ( q ; q ) k (6.31)which can be rewriting in terms of φ as η ( ∞ ) ( m, z, ξ ) = ( q − m λ ; q ) m ( y, αy ; q ) ∞ φ (cid:18) q − m , y, αyλq − m , (cid:12)(cid:12)(cid:12) q ; q (cid:19) (6.32)where y = iz r − qq m − = q − / e iκξ z r − qq m − iq / e − iκξ and αy = − iz r − qq m − e iκξ = q − / e iκξ z r − qq m − iq / e − iκξ . Next, recalling the definition of the Al-Salam-Chihara polynomials in (6.8) and taking into account theprevious prefactors, we arrive at the announced result in (6.7). (cid:3)
Appendix B
First, by using the identity (2.6), the r.h.s of (6.10) can be writing in the form B qm [ f ]( z ) = ( − m (cid:18) q m − m m [ m ] q ! √ π (cid:19) − / × Z R e imκξ − ξ ( iz q − qq m − , − iz q − qq m e iκξ ; q ) ∞ Q m ( τ s ; 2 τ α, τ β | q ) τ m f ( ξ ) dξ (6.33)where s = s q m − − q ie − iκξ q / − ie iκξ q − / √ , α = q − / z √ e iκξ , β = q / ¯ z √ e − iκξ and τ = r − q q m − . (6.34)11ext, we denote G q ( z ; ξ ) := 1 (cid:16) iz q − qq m − , − iz q − qq m e iκξ ; q (cid:17) ∞ . (6.35)Then by (2.3), it if follows successively thatLog G q ( z ; ξ ) = − X k ≥ Log (cid:16) izq − m p − q ( e iκξ − √ q ) q k + z (1 − q ) q − m e iκx q k (cid:17) = − izq − m p − q ( e iκξ − √ q ) X k ≥ q k − z (1 − q ) q − m e iκξ X k ≥ q k + o (1 − q )= izq − m ( √ q − e iκξ ) 1 √ − q − z q − m e iκξ
11 + q + o (1 − q ) . (6.36)On the other hand, the quantities izq − m ( √ q − e iκξ ) √ − q and z q − m e iκξ
11 + q tend respectively to √ ξz and 12 z as q → κ → . Therefor, when q →
1, we havelim q → G q ( z ; ξ ) = e √ ξz − z . To get the limit of the polynomial quantity in (6.33), we recall the limit given by Atakishiyeva andAtakishiyev ([37], p.3): lim q → τ − m Q m ( τ s ; 2 τ a, τ b | q ) = H m ( s − a − b ) (6.37)for the parameters in (6.34), we arrive atlim q → ( − m (cid:18) q m − m m [ m ] q ! √ π (cid:19) − / Q m ( τ s ; 2 τ α, τ β | q ) = ( − m (2 m m ! √ π ) − H m (cid:18) ξ − z + ¯ z √ (cid:19) . Finally, by grouping the obtained two limits, we arrive at the assertion in (6.13). (cid:3)
Appendix C
Since the functions Φ m,qj ( z ) form an orthonormal basis of the space A q,m ( C ), the corresponding repro-ducing kernel is given by K q,m ( z, w ) := ∞ X j =0 Φ q,mj ( z )Φ q,mj ( w ) . (6.38)By replacing the functions Φ q,mj ( z ) by their explicit expressions define in (4.2), We split this last sum intotwo parts as K q,m ( z, w ) = S m ( < ∞ ) ( z, w ; q ) + S m ( ∞ ) ( z, w ; q ) (6.39)where S m ( < ∞ ) ( z, w ; q ) = m − X j =0 ( q, q ; q ) m q ( j )(1 − q ) m − j ¯ z m − j w m − j ( q, q ; q ) m − j q mj ( q ; q ) m ( q ; q ) j P j (cid:0) (1 − q ) z ¯ z ; q m − j | q (cid:1) P j (cid:0) (1 − q ) w ¯ w ; q m − j | q (cid:1) − m − X j =0 ( q, q ; q ) j q ( m )(1 − q ) j − m z j − m ¯ w j − m ( q, q ; q ) j − m q mj ( q ; q ) m ( q ; q ) j P m (cid:0) (1 − q ) z ¯ z ; q j − m | q (cid:1) P m (cid:0) (1 − q ) w ¯ w ; q j − m | q (cid:1) , S m ( ∞ ) ( z, w ; q ) = X j ≥ ( q, q ; q ) j q ( m )(1 − q ) j − m z j − m ¯ w j − m ( q, q ; q ) j − m q mj ( q ; q ) m ( q ; q ) j P m (cid:0) (1 − q ) z ¯ z ; q j − m | q (cid:1) P m (cid:0) (1 − q ) w ¯ w ; q j − m | q (cid:1) )= q ( m )( q ; q ) m (cid:18) λ (cid:19) m X j ≥ ( q ; q ) j ( q, q ; q ) j − m (cid:18) λq m (cid:19) j P m (cid:0) α ; q j − m | q (cid:1) P m (cid:0) β ; q j − m | q (cid:1) , (6.40)where λ = (1 − q ) z ¯ w, α = (1 − q ) z ¯ z and β = (1 − q ) w ¯ w . By making use of the relation ([35], p.3): P n ( x ; q − N | q ) = x N ( − − N q N ( N +1 − n )2 ( q N +1 ; q ) n − N ( q − N ; q ) n P n − N ( x ; q N | q ) (6.41)for parameters N = j − m, n = j , x = α and x = β , we obtain that S q,m ( < ∞ ) ( λ, t ) = 0 . For the infinite sumin (6.40), let us rewrite the Wall polynomial as ([20], p.260): P n ( x ; a | q ) = ( x − ; q ) n ( aq ; q ) n ( − x ) n q − ( n ) φ (cid:18) q − n , xq − n (cid:12)(cid:12) q ; aq n +1 (cid:19) (6.42)where n = m, x = (1 − q ) λ and a = q j − m . Next, by using (6.24) and (6.26) respectively, Eq.(6.39)becomes K q,m ( z, w ) = q ( m )( q ; q ) m (cid:18) λ (cid:19) m ( q − m α ; q ) m ( q − m β ; q ) m S mq ( α ; β ) (6.43)where S mq ( α ; β ) := X j ≥ ( q ; q ) j ( q ; q ) j − m q j − m +1 ; q ) m (cid:18) λq m (cid:19) j φ (cid:18) q − m , αq − m (cid:12)(cid:12)(cid:12) q ; q j +1 (cid:19) φ (cid:18) q − m , βq − m (cid:12)(cid:12)(cid:12) q ; q j +1 (cid:19) . (6.44)Recalling the identity (6.26), it follows that ( q ; q ) j − m ( q j − m +1 ; q ) m = ( q ; q ) j , and with (2.9), we get S mq ( α ; β ) = X j ≥ ( q − m λ ) j ( q ; q ) j X k ≥ ( q − m ; q ) k ( αq − m ; q ) k ( q j +1 ) k ( q ; q ) k X l ≥ ( q − m ; q ) l ( βq − m ; q ) l ( q j +1 ) l ( q ; q ) l = X k,l ≥ ( q − m ; q ) k ( αq − m ; q ) k q k ( q ; q ) k ( q − m ; q ) l ( βq − m ; q ) l q l ( q ; q ) l X j ≥ ( q − m + k + l λ ) j ( q ; q ) j . (6.45)Now, by applying the q -binomial theorem ([20], p.17): X n ≥ a n ( q ; q ) n = 1( a ; q ) ∞ , | a | < , (6.46)for a = q − m + k + l λ , the r.h.s of (6.45) takes the form S mq ( α ; β ) = X k,l ≥ ( q − m ; q ) k ( αq − m ; q ) k q k ( q ; q ) k ( q − m ; q ) l ( βq − m ; q ) l q l ( q ; q ) l q − m + k + l λ ; q ) ∞ . (6.47)By applying the identity (2.3) to the factor 1( q − m + k + l λ ; q ) ∞ , (6.47) can be rewritten as S mq ( α ; β ) = 1( q − m λ ; q ) ∞ X k ≥ ( q − m ; q ) k ( αq − m ; q ) k q k ( q ; q ) k X l ≥ ( q − m ; q ) l ( q − m λ ; q ) k + l ( βq − m ; q ) l q l ( q ; q ) l . (6.48)13ext, by writing ( q − m λ ; q ) l + k = ( q − m λ ; q ) k ( q k − m λ ; q ) l , it follows that S mq ( α ; β ) = 1( q − m λ ; q ) ∞ X k ≥ ( q − m , q − m λ ; q ) k ( αq − m ; q ) k q k ( q ; q ) k X l ≥ ( q − m , q k − m λ ; q ) l ( βq − m ; q ) l q l ( q ; q ) l = 1( q − m λ ; q ) ∞ X k ≥ ( q − m , q − m λ ; q ) k ( αq − m ; q ) k q k ( q ; q ) k φ (cid:18) q − m , q k − m λβq − m (cid:12)(cid:12)(cid:12) q ; q (cid:19) , (6.49)in terms of the series φ . The latter one satisfies the identity ([15], p.10): φ (cid:18) q − n , bc (cid:12)(cid:12)(cid:12) q ; q (cid:19) = ( b − c ; q ) n ( c ; q ) n b n , n = 0 , , , ..., (6.50)which, with the parameters n = m , b = q k − m λ and c = βq − m , allows us to rewrite (6.49) as S mq ( α ; β ) = 1( q − m λ ; q ) ∞ X k ≥ ( q − m , q − m λ ; q ) k ( αq − m ; q ) k q k ( q ; q ) k ( q − k βλ ; q ) m ( q − m β ; q ) m ( q k − m λ ) m . (6.51)Furthermore, applying the identity( aq − n ; q ) k = ( a − q ; q ) n ( a − q − k ; q ) n ( a ; q ) k q − nk , a = 0 (6.52)to ( q − k βλ ; q ) m , gives that S mq ( α ; β ) = ( q − m λ ) m ( q wz ; q ) m ( q − m β ; q ) m ( q − m λ ; q ) ∞ X k ≥ ( q − m , q − m λ, zw ; q ) k ( q − m α, q − m zw ; q ) k q k ( q ; q ) k . (6.53)Finally, summarizing the above calculations, we arrive at K q,m ( z, w ) = q − m ( q − m α ; q ) m ( q wz ; q ) m ( q − m λ ; q ) ∞ ( q ; q ) m X k ≥ ( q − m , q − m λ, zw ; q ) k ( q − m α, q − m zw ; q ) k q k ( q ; q ) k = ( q − m α ; q ) m ( q wz ; q ) m q m ( q − m λ ; q ) ∞ ( q ; q ) m φ q − m , q − m λ, zwq − m α, q − m zw (cid:12)(cid:12)(cid:12) q ; q . (6.54)By making appeal to the finite Heine transformation ([38], p.2): φ (cid:18) q − n , ξ, βγ, q − n /τ (cid:12)(cid:12)(cid:12) q ; q (cid:19) = ( ξ τ ; q ) n ( τ ; q ) n φ (cid:18) q − n , γ/β, ξγ, ξ τ (cid:12)(cid:12)(cid:12) q ; β τ q n (cid:19) (6.55)for the parameters ξ = z/w, β = q − m λ, γ = q − m α, τ = qw/z , the reproducing kernel reduces to K q,m ( z, w ) = ( q − m (1 − q ) z ¯ z ; q ) m q m ( q − m (1 − q ) z ¯ w ; q ) ∞ φ q − m , zw , q ¯ z ¯ wq, q − m +1 (1 − q ) z ¯ z (cid:12)(cid:12)(cid:12) q ; q (1 − q ) w ¯ w ! . (6.56)For the limit, by taking into account the relation (2.7), we get thatlim q → ( q − m (1 − q ) z ¯ z ; q ) m q m ( q − m (1 − q ) z ¯ w ; q ) ∞ = lim q → q − m ( q − m (1 − q ) z ¯ z ; q ) m e q ( q − m z ¯ w ) = exp ( z ¯ w ) . (6.57)14y another side, using (2.9) together with the fact that ( q − m ; q ) k = 0 , ∀ k > m , the series φ in (6.15)terminates as m X k =0 ( q − m , z/w, q ¯ z/ ¯ w ; q ) k ( q − m (1 − q ) z ¯ z, q ; q ) k ( q (1 − q ) w ¯ w ) k ( q ; q ) k . 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