Generalized q-Laguerre type polynomials and q-partial differential equations

Abstract

In this paper we define the q-Laguerre type polynomials Un(x; y; z; q), which include q-Laguerre polynomials, generalized Stieltjes-Wigert polynomials, little q-Laguerre polynomials and q-Hermite polynomials as special cases. We also establish a generalized q-differential operator, with which we build the relations between analytic functions and Un(x; y; z; q) by using certain q-partial differential equations. Therefore, the corresponding conclusions about q-Laguerre polynomials, little q-Laguerre polynomials and q-Hermite polynomials are gained as corollaries. As applications, some generating functions and generalized Andrews-Askey integral formulas are given in the final section.