Dual Polynomials of the Multi-Indexed ($q$-)Racah Orthogonal Polynomials

We consider dual polynomials of the multi-indexed ($q$-)Racah orthogonal polynomials. The $M$-indexed ($q$-)Racah polynomials satisfy the second order difference equations and various $1+2L$ ($L\geq M+1$) term recurrence relations with constant coefficients. Therefore their dual polynomials satisfy the three term recurrence relations and various $2L$-th order difference equations. This means that the dual multi-indexed ($q$-)Racah polynomials are ordinary orthogonal polynomials and the Krall-type. We obtain new exactly solvable discrete quantum mechanics with real shifts, whose eigenvectors are described by the dual multi-indexed ($q$-)Racah polynomials. These quantum systems satisfy the closure relations, from which the creation/annihilation operators are obtained, but they are not shape invariant.


Introduction
Ordinary orthogonal polynomials in one variable satisfying second order differential equations are severely restricted by Bochner's theorem [1,2]. Allowed polynomials are the Hermite, Laguerre, Jacobi and Bessel polynomials, but the weight function of the Bessel polynomial is not positive definite. Various attempts to avoid this no-go theorem have been carried out and there are three directions.
The first direction (i) is to change the second order to higher orders. This direction was initiated by Krall [3] and he classified the orthogonal polynomials satisfying fourth order differential equations [4]. Based on the Laguerre and Jacobi polynomials, by adding the Dirac delta functions to the weight functions, orthogonal polynomials satisfying higher order differential equations are obtained [5]- [9]. Such polynomials are called the Krall polynomials. The second direction (ii) is to replace a differential equation with a difference equation. Studies in this direction were summarized as the Askey scheme of (basic-)hypergeometric orthogonal polynomials and various generalizations of the Bochner's theorem were proposed [10,2]. By combining (i) and (ii), orthogonal polynomials satisfying higher order difference equations were also studied. We call such polynomials the Krall-type polynomials. Some of them have weight functions with delta functions [11] and some others have those without delta functions [12]- [14]. The third direction (iii) is to allow missing degrees. This means the following situation: polynomials {P n } (n ∈ Z ≥0 ) are orthogonal and satisfy second order differential equation and form a complete set, but there are missing degrees, {deg P n |n ∈ Z ≥0 } Z ≥0 [15,16]. By combining with (ii), we can also consider the situation in which second order differential equation is replaced with second order difference equation. These polynomials are called exceptional or multi-indexed orthogonal polynomials and various examples have been obtained for classical orthogonal polynomials [15]- [33]. We distinguish the following two cases; the set of missing degrees I = Z ≥0 \{deg P n |n ∈ Z ≥0 } is case- (1) Quantum mechanical formulation is useful for studying orthogonal polynomials. We consider three kinds of quantum mechanical systems: ordinary quantum mechanics (oQM), discrete quantum mechanics with pure imaginary shifts (idQM) [34]- [37] and discrete quantum mechanics with real shifts (rdQM) [38]- [40]. Their features are the following: Schrödinger eq. variable x examples of orthogonal polynomials oQM differential eq. continuous Hermite, Laguerre, Jacobi idQM difference eq. continuous continuous Hahn, (Askey-)Wilson rdQM difference eq. discrete Hahn, (q-)Racah In our previous works we have taken second order differential or difference operators as Hamiltonians, but it is also allowed to take higher order operators as Hamiltonians. Exceptional and multi-indexed polynomials are obtained by applying the Darboux transformations with appropriate seed solutions to the exactly solvable quantum mechanical systems described by the classical orthogonal polynomials in the Askey scheme. When the virtual state wavefunctions are used as seed solutions, the case-(1) multi-indexed polynomials are obtained [21,23,25]. When the eigenstate and/or pseudo virtual state wavefunctions are used as seed solutions, the case-(2) multi-indexed polynomials are obtained [31]- [33]. Another method to obtain exceptional and multi-indexed polynomials is to use the Krall-type polynomials [28]- [30].
In this paper we discuss dual polynomials of the case-(1) multi-indexed (q-)Racah polynomials. Dual polynomials are introduced naturally for orthogonal polynomials of a discrete variable [2] and they are treated in the framework of rdQM [38]- [40]. The polynomial P n (η(x)) =P n (x) and its dual polynomial The roles of the variable and the label (= degree for ordinary orthogonal polynomials) are interchanged, and we have the following correspondence: The multi-indexed (q-)Racah polynomials satisfy the second order difference equations [25].
On the other hand, the multi-indexed polynomials do not satisfy the three term recurrence relations, which characterize the ordinary orthogonal polynomials [2], because they are not the ordinary orthogonal polynomials. They satisfy recurrence relations with more terms [41]- [48], and such recurrence relations for the multi-indexed (q-)Racah polynomials are studied recently [49]. It is shown that the M-indexed (q-)Racah polynomials satisfy various 1 + 2L (L ≥ M + 1) term recurrence relations with constant coefficients. Therefore dual polynomials of the multi-indexed (q-)Racah polynomials satisfy the three term recurrence relations and various 2L-th order difference equations, namely they are ordinary orthogonal polynomials and the Krall-type. The weight functions do not contain delta functions (Kronecker deltas). By using these dual multi-indexed (q-)Racah polynomials, we construct new exactly solvable rdQM systems, whose Hamiltonians are not tridiagonal but "(1 + 2L)-diagonal".
These quantum systems satisfy the closure relations [50,38], from which the creation and annihilation operators are obtained, but they are not shape invariant.
This paper is organized as follows. In section 2 the essence of the multi-indexed (q-)Racah polynomials are recapitulated. In section 3 we define the dual polynomials of the multiindexed (q-)Racah polynomials and present their properties. In section 4 we construct new exactly solvable rdQM systems described by the dual multi-indexed (q-)Racah polynomials.
The closure relations and the creation and annihilation operators are presented in § 4.1 and the shape invariance is discussed in § 4.2. Some examples are given in § 4.3. Section 5 is for a summary and comments. In Appendix A some basic data of the multi-indexed (q-)Racah polynomials are summarized for readers' convenience.
The Hamiltonian of the deformed system H D = (H D;x,y ) 0≤x,y≤xmax is a real symmetric matrix (a tridiagonal matrix in this case), where matrices e ±∂ are (e ±∂ ) x,y = δ x±1,y and the unit matrix 1 = (δ x,y ) are suppressed. The ), 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). Namely, it is a matrix product diag(f (0), f (1), . . . f (x max )) A diag(g(0), g(1), . . . , g(x max )). The notation Af (x) stands for a vector whose x-th component is which satisfy the boundary conditions The eigenvectors of the Hamiltonian are . . , n max ), (2.17) and the normalization of ψ D and φ D n is ψ D (0; λ) = φ D n (0; λ) = 1. Namely the multi-indexed (q-)Racah polynomials satisfy the second order difference equations H D (λ)P D,n (x; λ) = E n (λ)P D,n (x; λ) (n = 0, 1, . . . , n max ), (2.18) where the similarity transformed Hamiltonian The orthogonality relations of the multi-indexed (q-)Racah polynomials are xmax x=0 ψ D (x; λ) 2 Ξ D (1; λ)P D,n (x; λ)P D,m (x; λ) = δ nm d D,n (λ) 2 (n, m = 0, 1, . . . , n max ). (2.20) We remark that where A n and C n are the coefficients of the three term recurrence relations for the original (q-)Racah polynomials P n (η) andẼ v is the virtual state energy.

22)
where Ξ D Y means a polynomial (Ξ D Y )(η) = Ξ D (η)Y (η), and defineX(x) =X(x; λ) by Then the multi-indexed (q-)Racah polynomials P D,n (η) satisfy 1+2L term recurrence relations with constant coefficients: is an equation as a polynomial, namely it holds for x ∈ C. On the other hand, for We note that the overall normalization and the constant term of X(η) are not important, because the change of the former induces that of the overall normalization of r X,D n,k and the shift of the latter induces that of r X,D n,0 . The constant term of X(η) is chosen as X(0) = 0. There are the following relations among the coefficients r X,D n,k [49] Direct verification of this theorem is rather straightforward for lower M and smaller d j , n, deg Y and N, by a computer algebra system, e.g. Mathematica. The coefficients r X,D n,k are explicitly obtained for small d j and n. However, to obtain the closed expression of r X,D n,k for general n is not an easy task even for small d j , and it is a different kind of problem. Since Y (η) is arbitrary, we obtain infinitely many recurrence relations. Although not all of them are independent, the relations among them are unclear. Note that L ≥ M +1 because of ℓ D ≥ M.
The minimal degree one, which corresponds to

Dual Polynomials of the Multi-Indexed (q-)Racah Polynomials
For ordinary orthogonal polynomials, a discrete orthogonality relation of a system of polynomials induces an orthogonality relation for the dual system where the role of the variable and the degree are interchanged [2] (see also [38]- [40]). In this section we consider dual polynomials of the multi-indexed (q-)Racah polynomials, where the degree is replaced by the number of sign changes.
Corresponding to the multi-indexed (q-)Racah polynomialsP D,n (x; λ), let us defineQ D,x (n; λ) (Remark that if the parameter a is treated as an indeterminate,Q D,x (n; λ) is defined for Orthogonality relations (2.20) are rewritten as Since the matrix size is finite The second order difference equations forP D,n (x; λ) (2.18) are rewritten as the three term recurrence relations forQ D,x (n; λ), where we have used (2.10).
and satisfy the boundary conditions This means thatQ D,x (n; λ) are generated by the three term recurrence relations (3.7) with the initial conditionsQ These polynomials Q D,x (E) are ordinary orthogonal polynomials. So, the sign changes x times in the sequenceQ D, The multi-indexed (q-)Racah polynomials P D,n (η) and their dual polynomials Q D,x (E) are different polynomials. This contrasts with the original (q-)Racah cases. The (q-)Racah polynomials and their dual polynomials are same polynomials with the parameter correspondence We remark that if we treat the parameter a as an indeterminate, the dual multi-indexed (q-)Racah polynomials Q D,x (E) are defined for x ∈ Z ≥0 and E ∈ C by the three term recurrence relations rewritten as the 2L-th order difference equations forQ D,x (n; λ), (Remark: For L > 1 2 N, the order is not 2L but N.) Therefore the dual multi-indexed (q-)Racah polynomials Q D,x (E) are the Krall-type polynomials. Since we can take various X = X D,Y for a multi-indexed set D, these Krall-type polynomials Q D,x (E) satisfy various difference equations of order 2L ≥ 2M + 2.
In the q → 1 limit, the q-Racah polynomial reduced to the Racah polynomial [10]: Similarly, the (dual) multi-indexed q-Racah polynomials reduce to the (dual) multi-indexed Racah polynomials: lim

New Exactly Solvable rdQM Systems
In this section we present new exactly solvable rdQM systems, whose eigenvectors are described by the dual multi-indexed (q-)Racah polynomials. Unlike the previous section, we assume that the coordinate is x and the label of states is n as usual. Although tridiagonal matrices have been considered in our previous papers [38,39,40,33], the Hamiltonian of rdQM is not restricted to tridiagonal matrices and any real symmetric matrix is allowed.
We take a polynomial X(η) = X D,Y (η) (2.22) and assume that Y (η)( = 0) is a polynomial with real non-negative coefficients. For each X(η), we define the Hamiltonian H X dual where matrices e k∂ (k ∈ Z) are defined by , namely (e k∂ ) x,y = δ x+k,y .
This Hamiltonian H X dual D is real symmetric because of (2.26) (with n → x).
Therefore eigenvectors of the Hamiltonian H X dual are given by Their orthogonality relations are obtained from (3.6):   By the similarity transformation in terms of the ground state eigenvector φ dual D 0 (x), the Schrödinger equation (4.5) is rewritten as , we obtain the three term recurrence relations for the eigenvectors φ dual D n (x; λ), 0, 1, . . . , n max ; x = 0, 1, . . . , x max ). (4.14)

Closure relation
Corresponding to the three term recurrence relations (4.14), the Hamiltonian H X dual D and the sinusoidal coordinate E x are expected to satisfy the ordinary closure relation [50,38] (closure relation of order 2 [48]), For this purpose, we take (deg The method of closure relation [50,38] is the following: (i) Find R i (z) satisfying (4.15), annihilation operators a (±) are obtained. Here we change a part of the logic, namely mix (i) and (ii) by using some consequence of (iv). We define functions α ± (z) and polynomials R i (z) by guess work, which is expected from some consequence of (iv). Then we check the closure relation (4.15) for these R i (z).
From the closure relation (4.15), the exact Heisenberg operator solution of the sinusoidal coordinate E can be obtained [50,38], where a (±) = a (±) (H X dual Note that square roots in α ± (H X dual D ) are well-defined, because the matrix R 1 (H X dual is positive semi-definite, see (4.23). Action of (4.26) on φ dual D n (x) is On the other hand it turns out to be where we have used (4.14). Comparing these t-dependence, we obtain Remark: The value of functionX(x) (2.23) at x = −1 iš

No shape invariance
We will show that the rdQM system described by H X dual D (λ) is not shape invariant.
First let us factorize the Hamiltonian H X dual D (λ) (4.1). Since it is positive semi-definite, it can be factorized as where A X dual D (λ) is an upper triangular matrix (with upper bandwidth L for L ≤ 1 2 N). By imposing the condition A X dual D (λ) x,x ≥ 0 (x = 0, 1, . . . , x max ), this upper triangular matrix A X dual D (λ) = (a x,y ) 0≤x,y≤xmax is given by Here h ′ x,y are defined by where h x,y = H X dual D (λ) x,y and the convention n−1 i=n * = 0 is assumed. Note that the zero Next we recall the general theory of the shape invariance for finite rdQM systems (x max = n max = N) [38]. The Hamiltonian H(λ) = (H(λ) x,y ) 0≤x,y≤xmax is positive semi-definite, whose eigenvalues are 0 = E 0 (λ) < E 1 (λ) < · · · < E nmax (λ) and corresponding eigenvectors are φ n (x; λ), and factorized as H(λ) = A(λ) † A(λ), where A(λ) is upper triangular and A(λ) xmax,xmax = 0. Since the last row of A(λ) is zero, we have Let us write these N × N matrix B and N component vector b as (4.37) (In our previous studies [38]- [40], we treat tridiagonal Hamiltonians and A is an upper triangular matrix with upper bandwidth 1. Here this is not assumed.) Shape invariance is a relation between the system with parameters λ and that with λ ′ . Usually, appropriate choice of parameters allow us to express λ ′ as shifts of parameters λ ′ = λ + δ, but here we do not assume this. The number N, which corresponds to the size of the Hamiltonian, is one element of λ and it changes to N − 1 in λ ′ . Then the shape invariant condition is where κ is a positive constant and E 1 (λ) is the abbreviation for E 1 (λ)1 N . The Darboux transformation is defined by The shape invariant condition (4.38) gives namely, This relation implies that the energy eigenvalues E n (λ) are determined by the information of the first excited state energy E 1 (λ). For example, the energy eigenvalues E n (λ) for the original (q-)Racah systems are given by (2.8), and new set of parameters λ ′ is λ + δ. It is easy to check (4.42) and E n (λ) = n−1 s=0 κ s E 1 (λ + sδ) (n = 0, 1, . . . , N) for these cases. Now let us consider the shape invariance for the new exactly solvable rdQM H X dual D (λ). Assume that this system is shape invariant, Then (4.42) gives

Examples
To write down the Hamiltonian H X dual D (λ) (4.1) or similarity transformed one H X dual D (λ) (4.13), we need explicit form of the coefficients r X,D n,k (λ) (2.25). We can calculate r X,D n,k explicitly for small M, d j , deg Y and n, and check various properties of the system for small N: the Schrödinger equation (4.5) (or (4.12)), commutativity (4.11), the closure relation (4.15) (or but their explicit forms are somewhat lengthy. Here we write down r X,D n,k for D = {1} and Y = 1 [49]. For other cases, we present X(η) only. From X(η) = X(η; λ) and (2.23), the energy eigenvalues E X dual D,n (λ) (4.7) are obtained. Since the overall normalization of X(η) is not important, we multiply X(η) (2.22) by an appropriate positive factor.
r X,D n,−k + r X,D n,k .
r X,D n,−k + r X,D n,k .
We construct new exactly solvable rdQM systems, whose eigenvectors are described by the dual multi-indexed (q-)Racah polynomials. Their Hamiltonians (4.1) are not tridiagonal but "(1 + 2L)-diagonal". These quantum systems satisfy the closure relations (4.15), from which the creation/annihilation operators (4.29) are obtained, but they are not shape invariant. As a sufficient condition for exact solvability, we know two conditions: the closure relation and the shape invariance. Concerning the exactly solvable models we have studied, we observe that when they satisfy the (generalized) closure relation, they are also shape invariant. The new exactly solvable rdQM systems (4.1) give counterexamples to this observation.
Finally we list some problems related to the dual multi-indexed (q-)Racah polynomials.
1. The commutativity (4.11) originates from the non-uniqueness of X giving the recurrence relations with constant coefficients (2.25). The relations among the recurrence relations for various X (Y ) are unclear. It is an important problem to clarify them.
2. Orthogonal polynomials of a discrete variable in the Askey-scheme can be obtained as certain limits of the (q-)Racah polynomials [10]. It is an interesting problem to study various limits of the (dual) multi-indexed (q-)Racah polynomials. We remark that the (dual) multi-indexed (q-)Racah polynomials may not reduce to good polynomials in the same limits used for the (q-)Racah polynomials. For example, the case-(1) multiindexed polynomials are not allowed for some reduced polynomials. See [51] for similar situation in the Krall-type case.
3. For each (exactly solvable) rdQM system, we can construct the (exactly solvable) birth and death process [52], which is a stationary Markov chain. Acknowledgments I thank R. Sasaki for discussion and useful comments on the manuscript.

A Data for Multi-indexed (q-)Racah Polynomials
For readers' convenience, we present some data for the multi-indexed (q-)Racah polynomials [38,25,49], which are not presented in the main text.