A connection formula between the Ramanujan function and the $q$-Airy function

TL;DR

This paper establishes a connection formula linking the Ramanujan function and the $q$-Airy function using $q$-Borel and $q$-Laplace transformations, bridging two important $q$-special functions.

Contribution

It introduces a novel connection formula between two distinct $q$-Airy functions, expanding understanding of their relationship and applications.

Findings

Derived a connection formula between Ramanujan and $q$-Airy functions

Applied $q$-Borel and $q$-Laplace transformations in the derivation

Bridged functions from Ramanujan's work and $q$-Painlevé studies

Abstract

We show a connection formula between two different $q$-Airy functions. One is called the Ramanujan function which appears in Ramanujan's "Lost notebook". Another one is called the $q$-Airy function that obtained in the study of the second $q$-Painlev\'e equation. We use the $q$-Borel transformation and the $q$-Laplace transformation following C. Zhang to obtain the connection formula.