We construct (q, t)-Catalan polynomials and q-Fuss-Catalan polynomials for any irreducible complex reflection group W . The two main ingredients in this construction are Rouquier’s formulation of shift functors for the rational Cherednik algebras of W , and Opdam’s analysis of permutations of the irreducible representations of W arising from the Knizhnik-Zamolodchikov connection.

The aim of this paper is to construct generating functions for q-beta polynomials. By using these generating functions, we define the q -beta polynomials and also derive some fundamental properties of these polynomials. We give some functional equations and partial differential equations (PDEs) related to these generating functions. By using these equations, we find some identities related to t...

The q-classical polynomials are orthogonal polynomial sequences that are eigenfunctions of a second order q-differential operator of a certain type. We explicitly construct q-differential equations of arbitrary even order fulfilled by these polynomials, while giving explicit expressions for the integer composite powers of the aforementioned second order q-differential operator. The latter is ac...

We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-D Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the 2-D Hermite polynomials and a two variable extension of the Zernike or disc polynomials. We also give q-analogues of all these extensions. In each case in addit...

When τ is a quasi-definite moment functional onP , the vector space of all real polynomials, we consider a symmetric bilinear form φ(·, ·) on P ×P defined by φ(p, q) = λp(a)q(a)+ μp(b)q(b)+ 〈τ, p′q ′〉, where λ,μ, a, and b are real numbers. We first find a necessary and sufficient condition for φ(·, ·) to be quasi-definite. When τ is a semi-classical moment functional, we discuss algebraic prope...

The series solution is widely applied to differential equations onR but is not found in q-differential equations. Applying the Taylor andmultiplication rule of two generalized polynomials, we develop a series solution of linear homogeneous q-difference equations. As an example, the series solution method is used to find a series solution of the second-order q-difference equation of Hermite’s type.

This paper intends to define degenerate q-Hermite polynomials, namely polynomials by means of generating function. Some significant properties such as recurrence relations, explicit identities and differential equations are established. Many mathematicians have been studying the arising from functions special numbers polynomials. Based on results so far, we find for We also provide some using c...