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

Abstract

We define the $q$-Laguerre type polynomials $U_n(x,y,z;q)$, 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 relation between analytic functions  and $U_n(x,y,z;q)$ 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 obtained.