On Fractional <i>q</i>-Extensions of Some <i>q</i>-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.

Keywords

• <i>Q</i>-fractional calculus
• <i>Q</i>-hypergeometric functions
• <i>Q</i>-orthogonal polynomials
• fractional <i>q</i>-difference equations