相关论文: A second addition formula for continuous q-ultrasp…
The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling…
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…
Properties of the $q$-ultraspherical polynomials for $q$ being a primitive root of unity are derived using a formalism of the $so_q(3)$ algebra. The orthogonality condition for these polynomials provides a new class of trigonometric…
This article presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we…
L. Bary-Soroker and R. Shmueli (2026) have given an asymptotic formula for the number of irreducible polynomials over the finite fields $\mathbb F_q$ of $q$ elements, such that their coefficients are perfect squares in $\mathbb F_q$ and…
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…
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…
We introduce families of rational functions that are biorthogonal with respect to the $q$-hypergeometric distribution. A triplet of $q$-difference operators $X$, $Y$, $Z$ is shown to play a role analogous to the pair of bispectral operators…
The Al-Salam-Chihara polynomials are an important family of orthogonal polynomials in one variable $x$ depending on 3 parameters $\alpha$, $\beta$ and $q$. They are closely connected to a model from statistical mechanics called the…
We introduce two-parameter classes of exactly-solvable novel systems whose Hamiltonian operators could be represented by tridiagonal symmetric matrices in some orthogonal bases. The associated wavefunction is written as point-wise…
In this paper, we use the generalized q-polynomials with double q-binomial coefficients and homogeneous q-operators [J. Difference Equ. Appl. 20 (2014), 837--851.] to construct q-difference equations with seven variables, which generalize…
We recall five families of polynomials constituting a part of the so-called Askey-Wilson scheme. We do this to expose properties of the Askey-Wilson (AW) polynomials that constitute the last, most complicated element of this scheme. In…
The q-Legendre polynomials can be treated as some special "functions in the quantum double cosets $U(1)\setminus SU_q(2)/U(1)$". They form a family (depending on a parameter $q$) of polynomials in one variable. We get their further…
In this article, we exhaustively explore the terminating basic hypergeometric representations and transformations of the $q$ and $q^{-1}$-symmetric subfamilies of the Askey--Wilson polynomials. These subfamilies are obtained by repeatedly…
In this paper, we discuss new results related to the generalized discrete $q$-Hermite II polynomials $ \tilde h_{n,\alpha}(x;q)$, introduced by Mezlini et al. in 2014. Our aim is to give a continuous orthogonality relation, a $q$-integral…
Macdonald superpolynomials provide a remarkably rich generalization of the usual Macdonald polynomials. The starting point of this work is the observation of a previously unnoticed stability property of the Macdonald superpolynomials when…
Using a special case of Askey's $q$-beta integral evaluation formula, we determine orthogonality relations for the Al-Salam--Carlitz polynomials of type II with respect to a family of measures supported on a discrete subset of $\mathbb R$.…
In the present paper, new classes of wavelet functions are presented in the framework of Clifford analysis. Firstly, some classes of orthogonal polynomials are provided based on 2-parameters weight functions. Such classes englobe the well…
The $q$-analogue of the binomial coefficient, known as a $q$-binomial coefficient, is typically denoted $\left[{n \atop k}\right]_q$. These polynomials are important combinatorial objects, often appearing in generating functions related to…
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…