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 ${}_2{\varphi }_1(a,b;c;q,x)$. We obtain a list of seven different class of $q$-special functions: ${}_2{\varphi }_1,{}_1{\varphi }_1$, 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.

References in corpus (3)

• Lax forms of the $q$-Painlevé equations
• Hypergeometric solutions to the q-Painlevé equation of type $A_4^{(1)}$
• A connection formula between the Ramanujan function and the $q$-Airy function

Cited by in corpus (4)

• A connection formula between the Ramanujan function and the $q$-Airy function
• On local solutions of the Ramanujan equation and their connection formulae
• The Stokes phenomenon for the Ramanujan's $q$-difference equation and its higher order extension
• A Generalization of Zwegers' $\mu$-Function According to the $q$-Hermite-Weber Difference Equation