Casoratian Identities for the Discrete Orthogonal Polynomials in Discrete Quantum Mechanics with Real Shifts

In our previous papers, the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials and the Casoratian identities for the Askey-Wilson polynomial and its reduced form polynomials were presented. These identities are naturally derived through quantum mechanical formulation of the classical orthogonal polynomials; ordinary quantum mechanics for the former and discrete quantum mechanics with pure imaginary shifts for the latter. In this paper we present the corresponding identities for the discrete quantum mechanics with real shifts. Infinitely many Casoratian identities for the $q$-Racah polynomial and its reduced form polynomials are obtained.


Introduction
New types of orthogonal polynomials, the exceptional and multi-indexed orthogonal polynomials {P D,n (η)|n ∈ Z ≥0 }, have made remarkable progress in the theory of orthogonal polynomials and exactly solvable quantum mechanical models [1]- [27].Our approach to orthogonal polynomials is based on the quantum mechanical formulations: ordinary quantum mechanics (oQM), discrete quantum mechanics with pure imaginary shifts (idQM) [21]- [24] and discrete quantum mechanics with real shifts (rdQM) [25]- [27].For oQM, Schrödinger equations are second order differential equations.In discrete quantum mechanics, they are replaced by second order difference equations with a continuous variable for idQM or a discrete variable for rdQM.The Askey scheme of the (basic) hypergeometric orthogonal polynomials [28] is well matched to these quantum mechanical formulations: the Jacobi polynomial etc. in oQM, the Askey-Wilson polynomial etc. in idQM and the q-Racah polynomial etc. in rdQM.
From the exactly solvable quantum mechanical systems described by the classical orthogonal polynomials in the Askey scheme, we can obtain new exactly solvable quantum mechanical systems and various exceptional orthogonal polynomials with multi-indices by the multi-step Darboux transformations with appropriate seed solutions.One characteristic feature of these new types polynomials is the missing degrees.We distinguish the following two cases; the set of missing degrees I = Z ≥0 \{deg P n |n ∈ Z ≥0 } is case (1): I = {0, 1, . . ., ℓ − 1}, or case (2): I = {0, 1, . . ., ℓ − 1}, where ℓ is a positive integer.The situation of case ( 1) is called stable in [6].When the virtual state wavefunctions are used as seed solutions, the deformed systems are exactly iso-spectral to the original system and the case (1) multi-indexed orthogonal polynomials are obtained [8,10,12].When the eigenstate and/or pseudo virtual state wavefunctions are used as seed solutions, the deformed systems are almost iso-spectral to the original system but some states corresponding to the seed solutions are deleted or added, respectively, and the case (2) multi-indexed orthogonal polynomials are obtained [19,20].
For oQM and idQM, the deformed systems obtained by M-step Darboux transformations in terms of pseudo virtual state wavefunctions (with degrees specified by D) are equivalent to those obtained by multi-step Darboux transformations in terms of eigenstate wavefunctions (with degrees specified by D) with shifted parameters [19,20].The deformed system is characterized by the denominator polynomial Ξ D (η; λ) and the above equivalence is based on the proportionality of the denominator polynomials for each deformed system, Ξ D (η; λ) ∝ Ξ D(η; λ), (1.1) see [19,20] for details.Here we explain necessary notation only.Two sets D and D are defined for positive integers M and N (λ is a set of parameters and δ is its shift.): D def = {0, 1, . . ., N }\{ d1 , d2 , . . ., dM } def = {e 1 , e 2 , . . ., e N }.
(Exactly speaking, D and D should be treated as ordered sets.By changing the order of d j 's, Ξ D (η; λ) changes its overall sign.For the proportional relation (1.1), however, such an overall sign change does not matter.)The proportional relation (1.1) gives the Wronskian identity for oQM and the Casoratian identity for idQM.We write down the Casoratian identities for the Askey-Wilson polynomial: , Pe 2 , . . ., Pe N ](x; λ).
(1.3) (See Appendix A.1 for definitions of ξv (x; λ), Pn (x; λ) and ϕ M (x).)Here the Casorati determinant (Casoratian) of a set of n functions {f j (x)} for idQM is defined by (for n = 0, we set W γ [•](x) = 1) and γ = log q for the Askey-Wilson case.Based on the discrete symmetry of the system, the pseudo virtual state polynomial ξv (x; λ) is obtained from the eigenpolynomial Pn (x; λ) by twisting parameters.The Casoratian identities (1.3) represent the relation between Casoratians of the orthogonal polynomials with twisted and shifted parameters, and display the duality between the state adding and deleting Darboux transformations.The shape invariant properties of the original systems play an important role.
In this paper we consider the Casoratian identities for discrete orthogonal polynomials appearing in rdQM.A natural way to obtain them is the following; (a) Define the pseudo virtual state vectors by using discrete symmetries of the system, (b) Deform the system by multi-step Darboux transformations in terms of the pseudo virtual state vectors, (c) Compare it with the deformed system obtained by multi-step Darboux transformations in terms of the eigenvectors with shifted parameters.We present one-step Darboux transformation in terms of the pseudo virtual state vector by taking q-Racah case as an example in Appendix B.
However, calculation for multi-step cases is rather complicated.So, instead of the natural method mentioned above, we use a 'shortcut' method in this paper.We derive the Casoratian identities for the q-Racah polynomial (3.36) from those for the Askey-Wilson polynomial (1.3) by using the relation between the q-Racah and Askey-Wilson polynomials.The discrete orthogonal polynomials in the Askey scheme are obtained from the q-Racah polynomial in appropriate limits.The Casoratian identities for those reduced form polynomials can be obtained from those for the q-Racah polynomial in the same limits.
The Casoratian identities imply equivalences between the deformed systems obtained by multi-step Darboux transformations in terms of pseudo virtual state vectors and those in terms of eigenvectors with shifted parameters.For each (exactly solvable) rdQM system, we can construct the (exactly solvable) birth and death process [29], which is a stationary Markov chain.The Casoratian identities provide equivalence among such birth and death processes.
Similar Casoratian identities were studied by Curbera and Durán [30].Their method is different from ours and the identities are presented for the Charlier, Meixner, Krawtchouk and Hahn polynomials only, which have the sinusoidal coordinate η(x) = x.In our method various polynomials having five types of sinusoidal coordinates [25] This paper is organized as follows.In section 2 we recapitulate the discrete quantum mechanics with real shifts.Section 3 is the main part of this paper.After presenting the data for the original (q-)Racah systems in § 3.1, we discuss their discrete symmetries and present the pseudo virtual state polynomials by using the twist operations in § 3.2.The Casoratian identities for the (q-)Racah polynomials are derived starting from those for the Askey-Wilson polynomial in § 3.3.In section 4 the Casoratian identities for the reduced form polynomials are presented.Section 5 is for a summary and comments.In Appendix A some necessary data of orthogonal polynomials are presented.In Appendix B the pseudo virtual state vectors and one-step Darboux transformation are discussed by taking the q-Racah system as an example.

Discrete Quantum Mechanics with Real Shifts
In this section we recapitulate the discrete quantum mechanics with real shifts (rdQM) developed in [25,27].
The Hamiltonian of rdQM H = (H x,y ) is a tri-diagonal real symmetric (Jacobi) matrix and its rows and columns are indexed by integers x and y, which take values in {0, 1, . . ., N} (finite) or Z ≥0 (semi-infinite), The potential functions B(x) and D(x) are real and positive but vanish at the boundary, D(0) = 0 for both cases and B(N) = 0 for a finite case.In this paper we consider the case that these B(x) and D(x) are rational functions of x or q x (0 < q < 1).For simplicity in notation, we write the matrix H as follows: where matrices e ±∂ are e ±∂ = ((e ±∂ ) x,y ), (e ±∂ ) x,y and we suppress the unit matrix 1 = (δ x,y ) : B(x) + D(x) 1 in (2.2).The notation f (x)Ag(x), where f (x) and g(x) are functions of x and A is a matrix A = (A x,y ), stands for a matrix whose (x, y)-element is f (x)A x,y g(y).Note that the matrices e ∂ and e −∂ are not inverse to each other: e ±∂ e ∓∂ = 1 for a finite system and e −∂ e ∂ = 1 for a semi-infinite system.This Hamiltonian can be expressed in a factorized form: The Schrödinger equation is the eigenvalue problem for the hermitian matrix H, (n = 0, 1, . . ., N for a finite case).The ground state eigenvector φ 0 (x), which is characterized by Aφ 0 (x) = 0, is chosen as We use the convention: (2.7) Here P n (η) is a polynomial of degree n in η, and the sinusoidal coordinate η(x) is one of the following [25]: satisfy the boundary condition η(0) = 0. We adopt the universal normalization condition [25,27] as This Pn (x) is the eigenvector of the similarity transformed Hamiltonian H, H Pn (x) = E n Pn (x). (2.10) Explicitly, (2.10) is the difference equation for Pn , (2.11) Since P n is a polynomial, Pn (x) is defined for any x ∈ R and the difference equation (2.11) is also valid for x ∈ R. The eigenvectors are mutually orthogonal (d n > 0): where x max = N for a finite case, ∞ for a semi-infinite case.(Although this notation d n conflicts with the notation of the label of the pseudo virtual vector d j in (1.2), we think this does not cause any confusion because the former appears as 1 If we find functions B ′ (x) and D ′ (x) satisfying where α and α ′ are real constants, we obtain the following relation: where H ′ = (H ′ x,y ) 0≤x,y≤xmax is given by (2.16) For concrete examples, various quantities depend on a set of parameters λ = (λ 1 , λ 2 , . ..) and q.The parameter q is 0 < q < 1 and q λ stands for q (λ 1 ,λ 2 ,...) = (q λ 1 , q λ 2 , . ..).The The original systems in this paper are shape invariant [25] and they satisfy the relation, which is a sufficient condition for exact solvability.The auxiliary functions ϕ(x; λ) [25] and ϕ M (x; λ) [26] are defined by and ϕ 0 (x; λ) = ϕ 1 (x; λ) = 1.
For the orthogonal polynomials with Jackson integral measures such as the big q-Jacobi polynomial, the two component formulation is needed, see [27].
They are defined for a non-negative integer n by (a) n = n−1 j=0 (a + j) and (a; q) n = n−1 j=0 (1 − aq j ), which are extended to a real n by Note that, for a = 0 and n ∈ Z ≥0 , The hypergeometric series r F s and the basic hypergeometric series r φ s are where (a (for n = 0, we set W C [•](x) = 1).
For well-defined quantum systems, the range of parameters λ must be chosen such that the Hamiltonian is real symmetric.On the other hand, our main purpose of this paper is to obtain the Casoratian identities (3.36).Both sides of (3.36) are polynomials in x or Laurent polynomials in q x and (3.36) hold for any parameter range (except for the zeros of the denominators).So we do not bother about the range of parameters, except for Appendix B.
3 Casoratian Identities of the (q-)Racah Polynomials In this section we consider rdQM whose eigenvectors are described by the (q-)Racah polynomials.After discussing some discrete symmetries and pseudo virtual state polynomials, the Casoratian identities of the (q-)Racah polynomials are presented.

Discrete symmetries
Let us consider the twist operation t, which is an involution acting on x, λ and q, t(x, λ, q) = t(x), t(λ), t(q) , t 2 = id. (3.14) For R system, the q-part should be ignored.We present twist operations (i)-(ii) and ( i)-( ii), which lead to the pseudo virtual state vectors explained in Appendix B.
First let us define twist (ii) as follows: (ii) : t(x By using this twist operation, the functions B ′ (x) and D ′ (x) satisfying (2.13)-(2.14)are obtained: (ii) : ) We introduce the pseudo virtual state polynomial ξv (x; λ) which is the polynomial part of the pseudo virtual state vector, see Appendix B. It is a polynomial of degree v in η(x; λ), ξv (x; λ) By applying the symmetry (3.10) to this twist (ii), we define a twist (i): By using this twist (i), the functions B ′ (x) and D ′ (x) satisfying (2.13)-(2.14)are obtained: and α(λ) and α ′ (λ) are given by (3.17).The pseudo virtual state polynomial ξv (x; λ) which should be proportional to the twist (ii) case.In fact, we have and For qR case, we can change the parameter q.By applying the symmetry (3.11) to the twists (i)-(ii), we define the twists ( i)-( ii) for qR as follows: ( i) : (i) with the replacement t(q ( ii) : (ii) with the replacement t(q which give ( ii) : The pseudo virtual state polynomials ξv (x; λ) (v ∈ Z ≥0 ) are defined by and (3.13) implies ( i)-( ii) can be modified by exchanging t(λ 1 ) ↔ t(λ 2 ).

Casoratian identities
We will present the Casoratian identities for R and qR polynomials.By using discrete symmetries obtained in § 3.2, the pseudo virtual state vectors are defined, see Appendix B.
Then the original systems can be deformed by multi-step Darboux transformations in terms of pseudo virtual state vectors.The original systems can be also deformed by multi-step Darboux transformations in terms of eigenstate vectors [23].For appropriate choice of the index sets and shift of parameters, these two deformed systems are found to be equivalent.We present such calculation for one-step Darboux transformation in terms of the pseudo virtual state vector for q-Racah case in Appendix B. However, multi-step Darboux transformations in terms of pseudo virtual state vectors are rather complicated due to the following two facts: (a) the pseudo virtual state vectors do not satisfy the Schrödinger equation at both boundaries, (b) the size of the Hamiltonian increases at each step (namely the (N +1)×(N +1) matrix becomes the (N + M + 1) × (N + M + 1) matrix after M-step).So we present and prove the Casoratian identities for R and qR polynomials in a 'shortcut' way.
We take M, N , N , d j , dj , e j and λ as (1.2).Then the Casoratian identities for R and qR polynomials are and ϕ M (x; λ) are defined for real x, these identities hold for real x.For qR case, we prove (3.36) by translating the Casoratian identities (1.3) for the Askey-Wilson polynomial.The identities for R case is easily obtained from qR case by taking q → 1 limit.The necessary data of the Askey-Wilson polynomial are given in Appendix A.1.
The q-Racah polynomial and the Askey-Wilson polynomial are the 'same' polynomials [28].The replacement rule of this correspondence is ). (3.37) Under this replacement rule, we have and The shifts δ AW and δ qR are consistent.The twist t AW (λ AW ) (A.2) gives the twist (i) In the following we omit the superscript qR.The twisted potential function of AW system V ′ (A.3), the pseudo virtual state energy ẼAW v (A.4), the pseudo virtual state polynomial ξAW v (A.5) and the auxiliary function ϕ AW M (A.6) become The shifted parameters λAW and λ are also consistent.For functions f j (x AW ; λ AW ) = g j (x; λ), the Casoratian for idQM W γ (1.4) and that for rdQM W C (2.24) are related by is rewritten as By the replacement x → x − M +1 2 , this gives the Casoratian identities for qR polynomial (3.36).
Although the proportionality constants of (3.36) are not so important, we present them for the qR case.By explicit calculation (we assume where . Then the proportionality constants (we assume Here we have used

Casoratian Identities for the Reduced Case Polynomials
It is well known that the other members of the Askey scheme polynomials of a discrete variable can be obtained by reductions from the (q-)Racah polynomials [28].Not only the polynomials themselves but also the Hamiltonians are reduced in appropriate limits (Overall rescalings may be needed).Some of the twist operations t (i)-(ii), ( i)-( ii) of R and qR systems are inherited to the reduced systems.The twist operation t is (3.14) and the q-part should be ignored for non q-polynomials.The twists (i)-(ii) and ( i)-( ii) act on x and q as (i) : t(x) and t(λ) will be given for each polynomial.For non q-polynomials, the twists ( i)-( ii) are irrelevant.For infinite systems, the twists (i) and ( i) should be applied.By using the twist operation, the potential functions B ′ (x; λ) and D ′ (x; λ) are defined by (3.22) (or η(x; λ; q), ξv (x; λ; q) def = ξ v η(x; λ; q); λ; q ).The pseudo virtual state energies Ẽv (λ) are defined in (B.10)-(B.11)and they satisfy (B.12).Then the Casoratian identities for these reduced case polynomials have the same form as (3.36), with the notation (1.2).
The fundamental data for the reduced case polynomials are listed in Appendix A.2-A.3.
In the following we present twist operations and explicit forms of the pseudo virtual state polynomials.For the (q-)Hahn, dual (q-)Hahn, (q-)Krawtchouk and dual q-Krawtchouk cases, we have two twist operations.Two pseudo virtual state polynomial ξv (x; λ) obtained by these twists are proportional and two pairs of potential functions (B ′ (x; λ), D ′ (x; λ)) are also proportional.Therefore the pseudo virtual state vectors obtained by these twists are proportional and the corresponding Casoratian identities are identical.

Finite cases 4.1.1 Hahn (Ha)
We have two twist operations: The explicit forms of the pseudo virtual state polynomials are (ii) : ξv (x; λ) which are proportional, and

dual Hahn (dHa)
We have two twist operations: (ii) : t(λ The explicit forms of the pseudo virtual state polynomials are which are proportional, and

Krawtchouk (K)
We have two twist operations: (ii) : The explicit forms of the pseudo virtual state polynomials are which are proportional, and

Semi-infinite cases 4.2.1 Meixner (M)
We have one twist operation: The explicit form of the pseudo virtual state polynomials is

Summary and Comments
In addition to the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials in oQM [19] and the Casoratian identities for the Askey-Wilson polynomial and its reduced form polynomials in idQM [20], infinitely many Casoratian identities for the q-Racah polynomial and its reduced form polynomials in rdQM are obtained.The pseudo virtual state polynomials are defined by using discrete symmetries of the original systems.The derivation of the Casoratian identities in this paper is a 'shortcut' way.The pseudo virtual state vectors and the one-step Darboux transformation in terms of it for q-Racah case are discussed in Appendix B. We will report on the multi-step cases and semi-infinite cases elsewhere.The Casoratian identities imply equivalences between the deformed systems obtained by multistep Darboux transformations in terms of pseudo virtual state vectors and those in terms of eigenvectors with shifted parameters.Among the reduced form polynomials, the big q-Jacobi family and the discrete q-Hermite II are not mentioned in this paper.The orthogonality relations of the big q-Jacobi family are expressed in terms of the Jackson integral and their rdQM need two component formalism [27].The rdQM for the discrete q-Hermite II is an infinite system, x ∈ Z.We have not completed the study of the pseudo virtual state vectors for these two systems.It is plausible that similar Casoratian identities, which are polynomial identities, do exist.In fact R. Sasaki has checked tentative Casoratian identities for the big q-Jacobi family (private communication).
for which the Hamiltonian is well-defined, namely real symmetric, and the constant α(λ) is positive.Potential functions B ′ (x; λ) and D ′ (x; λ) are given by (3.23).We restrict the parameter range, ac < dq, b < q, d < q The range of v may be restricted.After (2.6), let us define φ0 (x; λ) by φ0 (x; λ) which is an 'almost' zero mode of H ′ (2.16), We define the pseudo virtual state vector φv (x; λ) as follows: This pseudo virtual state vector satisfies where the pseudo virtual state energy Ẽv (λ) is defined by (we present it for twists (ii) and ( i), ( ii) : Ẽv (λ; q) def = α(λ; q)E v t(λ); q −1 + α ′ (λ; q).(B.11) Namely the pseudo virtual state vector almost satisfies the Schrödinger equation, except for both boundaries x = 0 and x = N.This is in good contrast to the virtual state vector in rdQM [12], which fails to satisfy the Schrödinger equation at only one of the boundaries.
Let us introduce potential functions Bd 1 (x; λ) and Dd 1 (x; λ) determined by one of the pseudo virtual state polynomials ξd 1 (x; λ) : This establishes the equivalence of the two systems.

. 22 )
Since the inversion of the coordinate x (x → −x − 1) means the exchange of the matrices e ∂ ↔ e −∂ , the functions B(x) and D(x) are exchanged in this definition.Explicit forms of B ′ (x; λ) and D ′ (x; λ) are the same as (3.18), .35) The twist operations (i)-(ii) and ( i)-( ii) have essentially the same effects for R and qR systems, because the relations (3.23), (3.32), (3.26) and (3.35) imply that the pseudo virtual vectors (B.8) obtained by these twists are same (proportional).So they lead to the same Casoratian identities (3.36).However, they may have different effects for the reduced systems in § 4, which are obtained as appropriate limits of R and qR systems, because the symmetries (3.10)-(3.11)may no longer hold for the reduced systems.Since R and qR systems are invariant under the exchange a ↔ b, twists (i)-(ii) and .44) By using (3.38), (3.42)-(3.43)and (3.44), the Casoratian identities for AW polynomial (1.3) Curbera and Durán studied similar Casoratian identities for the Charlier, Meixner and Hahn polynomials [30], which have the sinusoidal coordinate η(x) = x.Their method is based on the Krall discrete measure.Let us consider the case N = max(D) and min(D) ≥ 1.Then their map I implies I(D) = D and I( D) = D, and we have max( D) = N and min( D) ≥ 1.Our identities (4.5) correspond to their Theorem 1.1, 5.1 and 7.1 as follows: Charlier: F = D, Meixner: F 1 = D and F 2 = ∅, Hahn: F 1 = D and F 2 = F 3 = ∅ (Remark: I(∅) = ∅ and max(∅) = −1).The proportionality constants are also presented.