Related papers: Matrix-Valued Little q-Jacobi Polynomials
In this paper, we consider a natural extension of several results related to Krall-type polynomials introducing a modification of a $q$-classical linear functional via the addition of one or two mass points. The limit relations between the…
A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given which can be considered as a limit formula for orthogonal polynomials. This is extended to a multi-parameter limit with 3 parameters, also involving (q-)Hahn…
It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a…
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…
Using the notion of quantum integers associated with a complex number $q\neq 0$, we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little $q$-Jacobi polynomials when $|q|<1$, and…
We provide an explicit expression for the first order $q$-difference system for the Jackson integral of symmetric Selberg type. The $q$-difference system gives a generalization of $q$-analog of contiguous relations for the Gauss…
We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal…
A new recurrence relation for exceptional orthogonal polynomials is proposed, which holds for type 1, 2 and 3. As concrete examples, the recurrence relations are given for Xj-Hermite, Laguerre and Jacobi polynomials in j = 1,2 case.
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…
Sequences of orthogonal polynomials that are alternative to the Jacobi polynomials on the interval $[0,1]$ are defined and their properties are established. An $(\alpha,\beta)$-parameterized system of orthogonal polynomials of the…
We consider multiple orthogonal polynomials corresponding to two Macdonald functions (modified Bessel functions of the second kind), with emphasis on the polynomials on the diagonal of the Hermite-Pad\'e table. We give some properties of…
This paper studies a new family of Angelesco multiple orthogonal polynomials with shared orthogonality conditions with respect to a system of weight functions, which are complex analogues of Pascal distributions on a legged star-like set.…
In this work we classify all the order-two Hypergeometric operators $D$, symmetric with respect to some $2\times 2$ irreducible matrix-weight $W$ such that $DP_n=P_n\left(\begin{smallmatrix} \lambda_n&0\\0&\mu_n \end{smallmatrix} \right)$…
A class of cross-shaped difference operators on a two dimensional lattice is introduced. The main feature of the operators in this class is that their formal eigenvectors consist of multiple orthogonal polynomials. In other words, this…
In this paper we study a generalization of the class of orthogonal polynomials on the real line. These polynomials satisfy the following relation: $(J_5 - \lambda J_3) \vec p(\lambda) = 0$, where $J_3$ is a Jacobi matrix and $J_5$ is a…
In this note we give a survey about polynomials whose moments are multiples of super Catalan numbers and explore two different kinds of q-analogues.
We define the analogue of Jack's (Jacobi) polynomials, which were defined for finite-dimensional root system by Heckman and Opdam as eigenfunctions of trigonometric Sutherland operator for the affine root system $\hat A_{n-1}$. In the…
In this article, the 2-iterated q-Appell family is introduced. Certain 2-iterated q-Appell and mixed type q-special polynomials are considered as members of this family. The numbers related to these polynomials are obtained. The determinant…
We construct the q-twisted cohomology associated with the q-multiplicative function of Jordan-Pochhammer type at |q|=1. In this framework, we prove the Heine's relations and a connection formula for the q-hypergeometric function of the…
The q-difference analog of the classical ladder operators is derived for those orthogonal polynomials arising from a class of indeterminate moments problem.