A unified approach to $q$-special functions of the Laplace type

Abstract

We propose a unified approach to $q$-special functions, which are degenerations of basic hypergeometric functions ${}_2\phi_1(a,b;c;q,x)$. We obtain a list of seven different class of $q$-special functions: ${}_2\phi_1, {}_1\phi_1$, two different types of the $q$-Bessel functions, the $q$-Hermite-Weber functions, two different types of the $q$-Airy functions. We show that there exist a relation between two types of the $q$-Airy functions.