On local solutions of the Ramanujan equation and their connection formulae

TL;DR

This paper derives connection formulae linking local solutions of the Ramanujan equation at the origin and infinity, utilizing q-Borel-Laplace transformations on functions like the Ramanujan function, q-Airy function, and divergent basic hypergeometric series.

Contribution

It introduces new connection formulae for the Ramanujan equation's solutions using two different q-Borel-Laplace transformations.

Findings

Connection formulae between solutions at origin and infinity

Application of q-Borel-Laplace transformations to hypergeometric series

Analysis of divergent basic hypergeometric series

Abstract

We show connection formulae of local solutions of the Ramanujan equation between the origin and the infinity. These solutions are given by the Ramanujan function, the $q$-Airy function and the divergent basic hypergeometric series ${}_2\varphi_0(0,0;-;q,x)$. We use two different $q$-Borel-Laplace transformations to obtain our connection formulae.