Related papers: Askey--Wilson polynomials, quadratic harnesses and…
An overview of the basic results on Macdonald(-Koornwinder) polynomials and double affine Hecke algebras is given. We develop the theory in such a way that it naturally encompasses all known cases. Among the basic properties of the…
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,…
For any orthogonal polynomials system on real line we construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian. The general scheme is divided into two types: a…
A tutorial introduction is given to q-special functions and to q-analogues of the classical orthogonal polynomials, up to the level of Askey-Wilson polynomials.
This semi-expository paper surveys results concerning three classes of orthogonal polynomials: in one non-hermitian variable, in several isometric non-commuting variables, and in several hermitian non-commuting variables. The emphasis is on…
We present a formula describing the asymptotics of a class of multivariate orthogonal polynomials with hyperoctahedral symmetry as the degree tends to infinity. The polynomials under consideration are characterized by a factorized weight…
The techniques for polynomial interpolation and Gaussian quadrature are generalized to matrix-valued functions. It is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the…
The weak convergence of orthogonal polynomials is given under conditions on the asymptotic behaviour of the coefficients in the three-term recurrence relation. The results generalize known results and are applied to several systems of…
We consider the exactly solvable quantum mechanical systems whose eigenfunctions are described by the multi-indexed orthogonal polynomials of Laguerre, Jacobi, Wilson and Askey-Wilson types. Corresponding to the recurrence relations with…
In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second--order linear partial differential equations, which are admissible potentially…
It has been known since 2007 that the Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere,…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated…
The Riemann-Hilbert problems for multiple orthogonal polynomials of types I and II are used to derive string equations associated to pairs of Lax-Orlov operators. A method for determining the quasiclassical limit of string equations in the…
We derive a generalized Rogers generating function and corresponding definite integral, for the continuous $q$-ultraspherical polynomials by applying its connection relation and utilizing orthogonality. Using a recent generalization of the…
This is a review of ($q$-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal…
We address the problem of constructing approximations based on orthogonal polynomials that preserve an arbitrary set of moments of a given function without loosing the spectral convergence property. To this aim, we compute the constrained…
In this survey we summarize the current state of known orthogonality relations for the $q$ and $q^{-1}$-symmetric and dual subfamilies of the Askey--Wilson polynomials in the $q$-Askey scheme. These polynomials are the continuous dual $q$…
Let $\{P_n \}_{n\ge0}$ be a sequence of monic orthogonal polynomials with respect to a quasi--definite linear functional $u$ and $\{Q_n \}_{n\ge0}$ a sequence of polynomials defined by $$Q_n(x)=P_n(x)+s_n P_{n-1}(x)+t_n P_{n-2}(x),\quad…
We study a limiting case of the Askey-Wilson polynomials when one of the parameters goes to infinity, namely continuous dual q-Hahn polynomials when q > 1. Solutions to the associated indeterminate moment problem by general theory are found…
We construct a pathwise calculus for functionals of integer-valued measures and use it to derive an martingale representation formula with respect to a large class of integer-valued random measures. Using these results, we extend the…