Related papers: Some Appell-type orthogonal polynomials on lattice…
The Askey--Wilson polynomials are the most general classical orthogonal polynomials that are known and the Nassrallah--Rahman integral is a very general extension of Euler's integral representation of the classical $_2F_1$ function. Based…
We provide an algebraic interpretation for two classes of continuous $q$-polynomials. Rogers' continuous $q$-Hermite polynomials and continuous $q$-ultraspherical polynomials are shown to realize, respectively, bases for representation…
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.
We study two families of orthogonal polynomials. The first is a finite family related to the Askey-Wilson polynomials but the orthogonality is on the real line. A limiting case of this family is an infinite system of orthogonal polynomials…
The idea of this review article is to discuss in a unified way the orthogonality of all positive definite polynomial solutions of the $q$-hypergeometric difference equation on the $q$-linear lattice by means of a qualitative analysis of the…
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…
The main purpose of this paper is to study generalized (self-) reciprocal Appell polynomials, which play a certain role in connection with Faulhaber-type polynomials. More precisely, we show for any Appell sequence when satisfying a…
In this paper, certain mixed special polynomial families associated with Appell sequences are introduced and their properties are established. Further, operational rules providing connections between these families and the known special…
We consider a quaternionic analogue of the univariate complex Hermite polynomials and study some of their analytic properties in some detail. We obtain their integral representation as well as the operational formulas of exponential and…
In the $q^{-1}$-symmetric Askey scheme, namely the $q^{-1}$-Askey--Wilson, continuous dual $q^{-1}$-Hahn, $q^{-1}$-Al-Salam--Chihara, continuous big $q^{-1}$-Hermite and continuous $q^{-1}$-Hermite polynomials, we compute bilateral discrete…
This manuscript contains a small portion of the algebraic theory of orthogonal polynomials developed by Maroni and their applicability to the study and characterization of the classical families, namely Hermite, Laguerre, Jacobi, and Bessel…
Some $q-$analogues of the normal ordering of the operator $(X+sD)^n$ on the polynomials are derived.
In this research, as the new results of our previously proposed definition for the new class of $2D$ $q$-Appell polynomials, we derive some interesting relations including the recurrence relation and partial $q$-difference equation of the…
We give equivalent forms of the Askey-Wilson polynomials expressing them with the help of the Al-Salam-Chihara polynomials. After restricting parameters of the Askey-Wilson polynomials to complex conjugate pairs we expand the Askey-Wilson…
In this chapter are given necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials. In addition, we also present the functional Rodrigues…
Recently there has been a renewed interest in an extension of the notion of orthogonal polynomials known as multiple orthogonal polynomials. This notion comes from simultaneous rational approximation (Hermite-Pade approximation) of a system…
We construct Lagrange interpolating polynomials for a set of points and values belonging to the algebra of real quaternions $H\simeq R_{0,2}$, or to the real Clifford algebra $R_{0,3}$. In the quaternionic case, the approach by means of…
In Lett. Math. Phys. 114, 54 (2024) and 115, 70 (2025), the author introduces what is presented as a novel method for determining whether a sequence of orthogonal polynomials is "classical", based solely on its initial recurrence…
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…
In this paper we propose a unified approach to matrix representations of different types of Appell polynomials. This approach is based on the creation matrix - a special matrix which has only the natural numbers as entries and is closely…