Orthogonal polynomials of a continuous variable in the Askey scheme satisfying second order difference equations, such as the Askey-Wilson polynomial, can be studied by the quantum mechanical formulation, idQM (discrete quantum mechanics with pure imaginary shifts). These idQM systems have the shape invariance property, which relates the Hilbert space H lambda (lambda: a set of parameters) and that with shifted parameters H lambda+delta (delta: shift of lambda), and gives the forward and backward shift relations for the orthogonal polynomials. Based on the forward shift relation and the Christoffel's theorem with some polynomial Phi(x), which is expressed in terms of the quantities appeared in the forward and backward shift relations, we obtain some difference relations for the orthogonal polynomials. The root multiplication of Phi(x) gives a surjective map from H lambda+2 delta to H lambda. Similarly, for the orthogonal polynomials in the Askey scheme satisfying second order differential equations, such as the Jacobi polynomial, we obtain some differential relations, and root the multiplication of Phi(x) in this case gives a surjective map from H lambda+delta to H lambda. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

For the isospectral Darboux transformations of the discrete quantum mechanics with real shifts, there are two methods: type I and type II constructions. Based on the type I construction, the type I multi-indexed little q-Jacobi and little q-Laguerre orthogonal polynomials were presented in Odake and Sasaki [J. Phys. A: Math. Theor. 50, 165204 (2017)]. Based on the type II construction, we present the type II multi-indexed little q-Jacobi and little q-Laguerre orthogonal polynomials.

We present the case-(1) multi-indexed orthogonal polynomials of a discrete variable for 8 types ((dual)(q-)Hahn, three kinds of q-Krawtchouk and q-Meixner). Based on them and the case-(1) multi-indexed orthogonal polynomials of Racah, q-Racah, Meixner, little q-Jacobi and little q-Laguerre types, exactly solvable continuous time birth and death processes are obtained. Their discrete time versions (Markov chains) are also obtained for finite types.

A new interpretation and applications of the 'Diophantine' and factorization properties of finite orthogonal polynomials in the Askey scheme are explored. The corresponding twelve polynomials are the (q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk. These (q-)hypergeometric polynomials, defined only for the degrees of $ 0,1,\ldots,N $ 0,1,& mldr;,N, constitute the main part of the eigenvectors of N + 1-dimensional tri-diagonal real symmetric matrices, which correspond to the difference equations governing the polynomials. The monic versions of these polynomials all exhibit the 'Diophantine' and factorization properties at higher degrees than N. This simply means that these higher degree polynomials are zero-norm 'eigenvectors' of the N + 1-dimensional tri-diagonal real symmetric matrices. A new type of multi-indexed orthogonal polynomials belonging to these twelve polynomials could be introduced by using the higher degree polynomials as the seed solutions of the multiple Darboux transformations for the corresponding matrix eigenvalue problems. The shape-invariance properties of the simplest type of the multi-indexed polynomials are demonstrated. The explicit transformation formulas are presented.

The forward and backward shift relations are basic properties of the (basic) hypergeometric orthogonal polynomials in the Askey scheme (Jacobi, Askey-Wilson, q-Racah, big q-Jacobi etc.) and they are related to the factorization of the differential or difference operators. Based on other factorizations, we obtain another type of forward and backward shift relations. Essentially, these shift relations shift only the parameters.

The discrete orthogonality relations for the multi-indexed orthogonal polynomials in discrete quantum mechanics with pure imaginary shifts are investigated. We show that the discrete orthogonality relations hold for the case-(1) multi-indexed orthogonal polynomials of continuous Hahn, Wilson, and Askey–Wilson types, and we conjecture their normalization constants.

The Hamiltonians of finite type discrete quantum mechanics with real shifts are real symmetric matrices of order N + 1. We discuss the Darboux transformations with higher degree (>N) polynomial solutions as seed solutions. They are state-adding and the resulting Hamiltonians after M-steps are of order N + M + 1. Based on twelve orthogonal polynomials ((q-)Racah, (dual, q-)Hahn, Krawtchouk and five types of q-Krawtchouk), new finite type multi-indexed orthogonal polynomials are obtained, which satisfy second order difference equations, and all the eigenvectors of the deformed Hamiltonian are described by them. We also present explicit forms of the Krein-Adler type multi-indexed orthogonal polynomials and their difference equations, which are obtained from the state-deleting Darboux transformations with lower degree (≤N) polynomial solutions as seed solutions.

About two dozens of exactly solvable Markov chains on one-dimensional finite and semi-infinite integer lattices are constructed in terms of convolutions of orthogonality measures of the Krawtchouk, Hahn, Meixner, Charlier, $q$-Hahn, $q$-Meixner and little $q$-Jacobi polynomials. By construction, the stationary probability distributions, the complete sets of eigenvalues and eigenvectors are provided by the polynomials and the orthogonality measures. An interesting property possessed by these stationary probability distributions, called `convolutional self-similarity,' is demonstrated.

We revisit the radial oscillator from the standpoint of a free oscillator realization. By using a free oscillator, namely, the creation/annihilation operators of the harmonic oscillator, we construct an operator that maps the eigenfunctions of the harmonic oscillator to those of the radial oscillator. As a polynomial part of this relation, we obtain an operator that maps the Hermite polynomials to the Laguerre polynomials.

Радиальный осциллятор рассматривается с точки зрения свободно-осцилляторной реализации. С использованием свободного осциллятора, а именно операторов рождения и уничтожения для гармонического осциллятора, построен оператор, который переводит собственные функции гармонического осциллятора в собственные функции радиального осциллятора. Как полиномиальная часть соотношения для собственных функций получается оператор, который переводит полиномы Эрмита в полиномы Лагерра.

Corresponding to a certain Wronskian identity, we present two types of new Casoratian identities. We apply these identities to the Darboux transformations of quantum mechanical systems. The Wronskian identity is applied to the ordinary quantum mechanics, and the two Casoratian identities are applied to the discrete quantum mechanics with pure imaginary and real shifts, respectively.

We review the recent development in the representation theory of the liVJ.+® algebra. The topics that we are concerned with are,

In previous papers, we discussed the recurrence relations of the multi-indexed orthogonal polynomials of the Laguerre, Jacobi, Wilson, Askey-Wilson, Racah and $q$-Racah types. In this paper we explore those of the Meixner-Pollaczek and continuous Hahn types. For the $M$-indexed Meixner-Pollaczek and continuous Hahn polynomials, we present $3+2M$ term recurrence relations with variable dependent coefficients and $1+2L$ term ($L\geq M+1$) recurrence relations with constant coefficients. Based on the latter, the generalized closure relations and the creation/annihilation operators of the quantum mechanical systems described by the multi-indexed Meixner-Pollaczek and continuous Hahn polynomials are obtained.

We present new exactly solvable systems of the discrete quantum mechanics with pure imaginary shifts, whose physical range of the coordinate is the whole real line. These systems are shape invariant and their eigenfunctions are described by the multi-indexed continuous Hahn and Meixner-Pollaczek orthogonal polynomials. The set of degrees of these multi-indexed polynomials are $\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}$, where $\ell_{\mathcal{D}}$ is an even positive integer ($\mathcal{D}$ : a multi-index set), but they form a complete set of orthogonal basis in the weighted Hilbert space.

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 >= M+1) term recurrence relations with constant coefficients. Therefore their dual polynomials satisfy the three term recurrence relations and various 2Lth-order difference equations. This means that the dual multi-indexed (q-)Racah polynomials are ordinary orthogonal polynomials of 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.

This is the second part of the project ‘unified theory of classical orthogonal polynomials of a discrete variable derived from the eigenvalue problems of hermitian matrices.’ In a previous paper, orthogonal polynomials having Jackson integral measures were not included, since such measures cannot be obtained from single infinite dimensional hermitian matrices. Here we show that Jackson integral measures for the polynomials of the big q-Jacobi family are the consequence of the recovery of self-adjointness of the unbounded Jacobi matrices governing the difference equations of these polynomials. The recovery of self-adjointness is achieved in an extended l2 Hilbert space on which a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a difference Schrödinger operator for an infinite dimensional eigenvalue problem. The polynomial appearing in the upper/lower end of Jackson integral constitutes the eigenvector of each of the two unbounded Jacobi matrix of the direct sum. We also point out that the orthogonal vectors involving the q-Meixner (q-Charlier) polynomials do not form a complete basis of the l2 Hilbert space, based on the fact that the dual q-Meixner polynomials introduced in a previous paper fail to satisfy the orthogonality relation. The complete set of eigenvectors involving the q-Meixner polynomials is obtained by constructing the duals of the dual q-Meixner polynomials which require the two component Hamiltonian formulation. An alternative solution method based on the closure relation, the Heisenberg operator solution, is applied to the polynomials of the big q-Jacobi family and their duals and q-Meixner (q-Charlier) polynomials.

A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of Hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schrödinger equations. The Hermitian matrices (factorizable Hamiltonians) are real symmetric tridiagonal (Jacobi) matrices corresponding to second order difference equations. By solving the eigenvalue problem in two different ways, the duality relation of the eigenpolynomials and their dual polynomials is explicitly established. Through the techniques of exact Heisenberg operator solution and shape invariance, various quantities, the two types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the coefficients of the three term recurrence, the normalization measures and the normalisation constants, etc., are determined explicitly.

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.

As the fourth stage of the project multi-indexed orthogonal polynomials, we present the multi-indexed Meixner and little q-Jacobi (Laguerre) polynomials in the framework of 'discrete quantum mechanics' with real shifts defined on the semi-infinite lattice in one dimension. They are obtained, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier, from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum–Krein–Adler deletion of virtual state vectors. The virtual state vectors are the solutions of the matrix Schrödinger equation on all the lattice points having negative energies and infinite norm. This is in good contrast to the (q-)Racah systems defined on a finite lattice, in which the 'virtual state' vectors satisfy the matrix Schrödinger equation except for one of the two boundary points.

The multi-indexed Laguerre and Jacobi polynomials form a complete set of orthogonal polynomials. They satisfy second-order differential equations but not three term recurrence relations, because of the 'holes' in their degrees. The multi-indexed Laguerre and Jacobi polynomials have Wronskian expressions originating from multiple Darboux transformations. For the ease of applications, two different forms of simplified expressions of the multi-indexed Laguerre and Jacobi polynomials are derived based on various identities. The parity transformation property of the multi-indexed Jacobi polynomials is derived based on that of the Jacobi polynomial.