Related papers: A Note on the Rogers-Szeg\"{o} Polynomial $q$-Diff…
In this paper, we introduce the deformed homogeneous polynomials $\mathrm{R}_{n}(x,y;u|q)$. These polynomials generalize some classical polynomials: the Rogers-Szeg\"o polynomials $\mathrm{h}_{n}(x|q)$, the generalized Rogers-Szeg\"o…
In this paper, by making use of the familiar $q$-difference operators $D_q$ and $D_{q^{-1}}$, we first introduce two homogeneous $q$-difference operators $\mathbb{T}({\bf a},{\bf b},cD_q)$ and $\mathbb{E}({\bf a},{\bf b}, cD_{q^{-1}})$,…
In this paper, we introduce the Heine binomial operators H$_{n}(bD_{q})$ based on $q$-differential operator $D_{q}$. The motivation for introducing the operators H$_{n}(bD_{q})$ is that their limit turns out to be the $q$-exponential…
We present an operator approach to Rogers-type formulas and Mehler's formulas for the Al-Salam-Carlitz polynomials $U_n(x,y,a;q)$. By using the q-exponential operator, we obtain a Rogers-type formula which leads to a linearization formula.…
In this paper, we deduce the generalized $q$-difference equations for general Al-Salam--Carlitz polynomials and generalize Arjika's recently results [$q$-difference equation for homogeneous $q$-difference operators and their applications,…
Let $(p_n)_n$ be either the $q$-Meixner or the $q$-Laguerre polynomials. We form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of two consecutive $p_n$: $q_n=p_n+\beta_np_{n-1}$, $\beta_n\in \RR$. Using the…
This paper addresses a new characterization of $({\cal R},p,q)-$deformed Rogers-Szeg\"o polynomials by providing their three-term recurrence relation and the associated quantum algebra built with corresponding creation and annihilation…
In this paper, we use the Rogers-Ramanujan type $q$-exponential operator $\mathcal{R}(qD_{q})$ to derive generating functions, and Mehler and Rogers formulas, for the non-normalized homogeneous Stieljes-Wigert polynomials…
The ring $\text{Diff}_{\mathbf{h}}(n)$ of $\mathbf{h}$-deformed differential operators appears in the theory of reduction algebras. In this thesis, we construct the rings of generalized differential operators on the $\mathbf{h}$-deformed…
We present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szeg\"{o} polynomials $h_n(x,y|q)$. The proof of Mehler's formula can be considered as a new approach to the nonsymmetric Poisson…
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…
We construct the rings of generalized differential operators on the ${\bf h}$-deformed vector space of ${\bf gl}$-type. In contrast to the $q$-deformed vector space, where the ring of differential operators is unique up to an isomorphism,…
Using the theory of functions of several complex variables, we prove that if an analytic function in several variables satisfies a system of $q$-partial differential equations, then, it can be expanded in terms of the product of the…
The Al-Salam & Carlitz polynomials are $q$-generalizations of the classical Hermite polynomials. Multivariable generalizations of these polynomials are introduced via a generating function involving a multivariable hypergeometric function…
We introduce a q-deformation of Dirichlet series : for each s, an operator acting on formal power series in q without constant term. We relate Bernoulli-Carlitz numbers to the q-Riemann Zeta operators for negative integers, evaluated on…
In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group $W$ of type $B_n$. We endow the polynomial ring ${\mathbb C} [x_1,\ldots\\\ldots, x_n]$ with a structure of…
We introduce the concept of $\D$-operators associated to a sequence of polynomials $(p_n)_n$ and an algebra $\A$ of operators acting in the linear space of polynomials. In this paper, we show that this concept is a powerful tool to generate…
We introduce, characterise and provide a combinatorial interpretation for the so-called $q$-Jacobi-Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order $q$-differential…
In this paper, we first construct the homogeneous $q$-shift operator $\widetilde{E}(a,b;D_{q})$ and the homogeneous $q$-difference operator $\widetilde{L}(a,b; \theta_{xy})$. We then apply these operators in order to represent and…
We construct a commutative algebra A_z, generated by d algebraically independent q-difference operators acting on variables z_1, z_2,..., z_d, which is diagonalized by the multivariable Askey-Wilson polynomials P_n(z) considered by Gasper…