New Determinant Expressions of the Multi-indexed Orthogonal Polynomials in Discrete Quantum Mechanics

The multi-indexed orthogonal polynomials (the Meixner, little $q$-Jacobi (Laguerre), ($q$-)Racah, Wilson, Askey-Wilson types) satisfying second order difference equations were constructed in discrete quantum mechanics. They are polynomials in the sinusoidal coordinates $\eta(x)$ ($x$ is the coordinate of quantum system) and expressed in terms of the Casorati determinants whose matrix elements are functions of $x$ at various points. By using shape invariance properties, we derive various equivalent determinant expressions, especially those whose matrix elements are functions of the same point $x$. Except for the ($q$-)Racah case, they can be expressed in terms of $\eta$ only, without explicit $x$-dependence.

The eigenstates of the deformed system are expressed in terms of determinants: Wronskians for ordinary quantum mechanics (in which the Schrödinger equation is a second-order differential PTEP 2017, 053A01 S. Odake
The parameters of the systems are and we adopt the following choice of the parameter ranges for R and qR: We list the fundamental data (Ref. [43]).
• Little q-Laguerre and little q-Jacobi: where p n (y; a|q) and p n (y; a, b|q) are the little q-Laguerre and little q-Jacobi polynomials in the conventional definition (Ref. [45]), respectively.

Virtual states
The virtual state vectors are obtained by using the discrete symmetries of the Hamiltonian (Refs. [12,14]). We define the twist operation t as which is an involution t 2 = id and satisfies By using the twist operation, the virtual state vectorsφ v (x; λ) are introduced: The virtual state polynomials ξ v (η; λ) are polynomials of degree v in η. The virtual state vectors satisfy the Schrödinger equation  The other data α(λ) and α (λ) (see Refs. [12,14] for details) are (2.41) In order to obtain well-defined quantum systems, we have to restrict the degree v and parameters λ so that the conditionsẼ v (λ) < 0, α(λ) > 0, α (λ) < 0, no zeros of ξ v (η; λ) in the domain, etc. are satisfied; see Refs. [12,14]. In this paper, however, we consider algebraic properties only and we do not bother about the ranges of v and λ.
The function ν(x; λ) is defined as the ratio of φ 0 andφ 0 : Since the potential functions B(x; λ) and D(x; λ) are rational functions of x or q x , this function ν(x; λ) can be analytically continued into a meromorphic function of x or q x through the functional relations Explicitly it is For a non-negative integer x, it reduces to The functions r j (x; λ, M ) are defined as the ratio of ν's: whose explicit forms are The auxiliary function ϕ M (x) (Ref. [35]) is defined by (2.48) and ϕ 0 (x) = ϕ 1 (x) = 1. For M, lqL, and lqJ, its explicit form is In the following we adopt the conventioň

Shape invariance
The original systems in Sect. 2.1 are shape invariant (Ref. [43]) and they satisfy the relation which is a sufficient condition for exact solvability. As a consequence of the shape invariance, the action of the operators A(λ) and A(λ) † on the eigenvectors is The relations in Eq. (2.51) are equivalent to the forward and backward shift relations of the orthogonal polynomials P n (Ref. [45]), where the forward and backward shift operators F(λ) and B(λ) are defined by  The actions of F(λ) and (For a finite system, the first equation holds for x ≤ N − 1 as a matrix and vector equation.) Explicitly they are equivalent to the following forward and backward shift relations of the virtual state polynomialξ v : and Note that the relations (2.57) (after writing down in components) and (2.58)-(2.59) are valid for any x ∈ R becauseξ v (x) is a "polynomial" and ν(x) is a meromorphic function.  [12,14]): Here the universal normalizationsˇ D (0; λ) = 1 andP D,n (0; λ) = 1 are adopted, which determines the constants C D (λ) and C D,n (λ) (convention: 1≤j<k≤M * = 1 for M = 1), The denominator polynomial D (η; λ) and the multi-indexed orthogonal polynomial P D,n (η; λ) in Eq. (2.65) are polynomials in η and their degrees are generically D and D + n, respectively. Here 10 In Ref. [12], the multi-indexed orthogonal polynomialsP D,n (x; λ) in Eq. (2.64) for R and qR are expressed aš . This expression is also valid for M, lqL, and lqJ.
In the next subsection we will rewrite Eqs. (2.63)-(2.64) using the identities implied by the shape invariance and the properties of the Casoratians.

New determinant expressions
(2.70) In the rest of this section we consider the Casoratians forP n (x),ξ v (x), and ν(x). Since they are polynomials or meromorphic functions, we can realize the shift operator as an exponential of the differential operator, exp a d dx f (x) = f (x + a) (a ∈ R). In contrast to the matrices e ∂ and e −∂ , the operators exp d dx and exp − d dx are inverse to each other. The following determinant formula holds for any smooth functions q j (x) and r j (x) (j = 1, 2, . . .): and the ordered product is n ←− j=1 a j def = a n · · · a 2 a 1 .
This formula is shown by using the properties of the determinants (row properties) and induction on n.

Multi-indexed orthogonal polynomials in idQM
In this section we derive various equivalent expressions for the multi-indexed Wilson and Askey-Wilson polynomials in the framework of discrete quantum mechanics with pure imaginary shifts.

Original systems
The Hamiltonian of discrete quantum mechanics with pure imaginary shifts (idQM) (Refs. [46,47]), whose dynamical variables are the real coordinate x (x 1 ≤ x ≤ x 2 ) and the conjugate momentum PTEP 2017, 053A01 S. Odake Here the potential function V (x) is an analytic function of x, and γ is a real constant. The * -operation on an analytic function f (x) = n a n x n (a n ∈ C) is defined by f * (x) = n a * n x n , in which a * n is the complex conjugation of a n . Since the momentum operator appears in exponentiated forms, the Schrödinger equation is an analytic difference equation with pure imaginary shifts. We consider idQM described by the Wilson and Askey-Wilson polynomials. The various parameters are where 0 < q < 1. The parameters are restricted by Here are the fundamental data (Ref. [46]):   a 2 , a 1 + a 3 , a 1 + a 4 1 : W, a 1 a 2 , a 1 a 3 , a 1 a 4 ; q) n a 1 a 2 , a 1 a 3 , a 1   : AW, (3.12) where W n (η; a 1 , a 2 , a 3 , a 4 ) and p n (η; a 1 , a 2 , a 3 , a 4 |q) are the Wilson and Askey-Wilson polynomials (Ref. [45]), respectively. Note that (3.14)

Virtual states
The virtual state wave functions are obtained by using the discrete symmetries of the Hamiltonian (Ref. [10]). In the following we restrict the parameters as follows: We define two twist operations t, type I and type II: (3.16) which are involutions t 2 = id and satisfy By using these two types of twist operations, two types of virtual state wave functions,φ I v (x; λ) and φ II v (x; λ), are introduced: The virtual state polynomials ξ v (η; λ) (two types ξ I v and ξ II v ) are polynomials of degree v in η. The virtual state wave functions satisfy the Schrödinger equation The other data α(λ) and α (λ) (see Ref. [10] for details) are In order to obtain well-defined quantum systems, we have to restrict the degree v and the parameters a i so that the conditions,Ẽ v (λ) < 0, α(λ) > 0, α (λ) < 0, no zeros of ξ v (η; λ) in the domain, etc., are satisfied; see Ref. [10]. In this paper, however, we consider algebraic properties only and we do not bother about the ranges of v and a i . The functions ν(x; λ) (two types ν I and ν II ) are defined as the ratio of φ 0 andφ 0 : The functions r j (x (M ) j ; λ, M ) (two types r I j and r II j ) are defined as the ratio of ν's: The auxiliary function ϕ M (x) (Ref. [34]) is defined by where the functionsǓ (x; λ) (two typesǓ I andǓ II ) arě In fact, these functionsǓ (x; λ) are polynomials U (η; λ) in η as defined by ; λ), (3.28) In the following we adopt the conventioň

Shape invariance
The original systems in Sect. 3.1 are shape invariant (Refs. [46,47]) and they satisfy the relation which is a sufficient condition for exact solvability. As a consequence of the shape invariance, the action of the operators A(λ) and A(λ) † on the eigenfunctions is ; λ), (3.32) where the factors of the energy eigenvalue, f n (λ) and b n−1 (λ), E n (λ) = f n (λ)b n−1 (λ), are given by The relations (3.32) are equivalent to the forward and backward shift relations of the orthogonal polynomials P n (Ref. [45]), ; λ), (3.34) where the forward and backward shift operators F(λ) and B(λ) are defined by Note that the forward shift operator F(λ) is parameter independent. The second-order difference operator H(λ) acting on the polynomial eigenfunctions is square-root-free. It is defined by ; λ). (3.38) The action of the operators A(λ) and A(λ) † on the virtual state wave functionsφ v (x; λ) (φ I v and φ II v ) is ; λ), (3.39) where the factors of the virtual energy eigenvalue, The relations (3.39) are equivalent to the forward and backward shift relations of (ν ; λ). (3.41) Explicitly they give square-root-free relations: forward and backward shift relations of the virtual polynomials ξ v (ξ I v and ξ II v ; Ref. [10]), ; λ), (3.42) where the functions v 1 (x; λ) and v 2 (x; λ) are   The following determinant formula holds for any analytic functions q j (x), r j (x), q j (x) and r j (x) (j = 1, 2, . . .): where the operatorsD j ,D j and functions D j (x) (j = 1, 2, . . .) arê This formula can be proven by using the properties of the determinant (row properties) and induction on n (even n → odd n + 1, odd n → even n + 1). By taking F or B asD j andD j and using the shape-invariance properties (3.34) and ( which correspond to cases A and B in Ref. [42]. For the function D j (x) in Eq. (3.57), we define a functionŠ(x; λ) and a polynomial S(η; λ): 1 a 2 a 3 + a 1 a 2 a 4 + a 1 a 3 a 4 + a 2 a 3 The shape-invariance properties (3.34) and (3.41) give (k = 0, 1, . . .) We rewrite Eqs.

Single type
As remarked after Eq. (3.51), when the index set D consists of type I indices only (or type II indices only), simplifications occur. We consider such a single-type index set here. As shown in Refs. [48,49], the multi-indexed polynomials with both types of indices can always be recovered from those with single-type indices D (type I only or type II only) with shifted parameters λ , namely, P D,n (η; λ) = c P D,n (λ) c P D ,n (λ ) P D ,n (η; λ ), (3.76) where c P D,n (λ) is the coefficient of the highest-degree term of P D,n (η; λ) (Ref. [10]), and D and λ are determined by D and λ (Ref. [48]).