相关论文: Duality of q-polynomials, orthogonal on countable …
In this manuscript we study algebraic and analytic properties of the sequence of monic polynomials orthogonal with respect to a Jacobi differential operator. A fluid dynamics model for source points location of a flow of an incompressible…
We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via…
A family of tridiagonal pairs which appear in the context of quantum integrable systems is studied in details. The corresponding eigenvalue sequences, eigenspaces and the block tridiagonal structure of their matrix realizations with respect…
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 Schr\"odinger…
Several kinds of q-orthogonal polynomials with |q|=1 are constructed as the main parts of the eigenfunctions of new solvable discrete quantum mechanical systems. Their orthogonality weight functions consist of quantum dilogarithm functions,…
We introduce a new family of special functions, namely $q$-Charlier multiple orthogonal polynomials. These polynomials are orthogonal with respect to $q$-analogues of Poisson distributions. We focus our attention on their structural…
A general theory of matrix-spherical functions for dual Hopf algebras and right coideal subalgebras is developed. We establish their existence and define their orthogonality relations. When specialized to Kolb and Letzter's quantum…
In this paper we study the orthogonality conditions satisfied by the classical q-orthogonal polynomials that are located at the top of the q-Hahn tableau (big q-jacobi polynomials (bqJ)) and the Nikiforov-Uvarov tableau (Askey-Wilson…
We introduce two q-analogues of the 2D-Hermite polynomials which are functions of two complex variables. We derive explicit formulas, orthogonality relations, raising and lowering operator relations, generating functions, and Rodrigues…
Following Verde-Star, Linear Algebra Appl. 627 (2021), we label families of orthogonal polynomials in the $q$-Askey scheme together with their $q$-hypergeometric representations by three sequences $x_k, h_k, g_k$ of Laurent polynomials in…
This paper discusses operators lowering or raising the degree but preserving the parameters of special orthogonal polynomials. Results for one-variable classical (q-)orthogonal polynomials are surveyed. For Jacobi polynomials associated…
Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We…
Let $d\nu$ be a measure in $\mathbb{R}^d$ obtained from adding a set of mass points to another measure $d\mu$. Orthogonal polynomials in several variables associated with $d\nu$ can be explicitly expressed in terms of orthogonal polynomials…
Following the pioneering work of Duistermaat and Gr\"unbaum, we call a family $\{p_n(x)\}_{n=0}^{\infty}$ of polynomials bispectral, if the polynomials are simultaneously eigenfunctions of two commutative algebras of operators: one…
For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal…
We investigate the differential equation for the Jacobi-type polynomials which are orthogonal on the interval $[-1,1]$ with respect to the classical Jacobi measure and an additional point mass at one endpoint. This scale of higher-order…
We investigate the distribution of zeros of the little q-Jacobi polynomials and related q-hypergeometric families. We prove that the zeros of these orthogonal polynomials exhibit strong interlacing properties and obey natural monotonicity…
The concept of cyclic tridiagonal pairs is introduced, and explicit examples are given. For a fairly general class of cyclic tridiagonal pairs with cyclicity N, we associate a pair of `divided polynomials'. The properties of this pair…
An algebraic interpretation of the one-variable quantum $q$-Krawtchouk polynomials is provided in the framework of the Schwinger realization of $\mathcal{U}_{q}(sl_{2})$ involving two independent $q$-oscillators. The polynomials are shown…
We establish an integral representations of a right inverses of the Askey-Wilson finite difference operator in an $L^2$ space weighted by the weight function of the continuous $q$-Jacobi polynomials. We characterize the eigenvalues of this…