On Fractional q-Extensions of Some q-Orthogonal Polynomials

Abstract

In this paper, we introduce a fractional q-extension of the q-differential operator Dq−1 and prove some of its main properties. Next, fractional q-extensions of some classical q-orthogonal polynomials are introduced and some of the main properties of the newly-defined functions are given. Finally, a fractional q-difference equation of Gaussian type is introduced and solved by means of the power series method.

A Unified Representation of q- and h-Integrals and Consequences in Inequalities

“…Roughly speaking, q-calculus analyzes q-analogues of mathematical concepts and formulas that can be recaptured by the limit q → 1. The concepts of q-calculus are extensively applied in various subjects of physics and mathematics, including combinatorics, number theory, orthogonal polynomials, geometric function theory, quantum theory and mechanics, and the theory of relativity; see [3][4][5][6][7][8][9].…”