The Stokes phenomenon for the Ramanujan's $q$-difference equation and its higher order extension

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 ${}_2{\phi }_0(0,0;-;q,x)$. We use two different $q$-Borel-Laplace resummation methods to obtain our connection formulae. We also introduce the $q$-Borel-Laplace transformation of level $r-1$, which are higher order extension of these transformations. These methods are useful to obtain an asymptotic formula of a divergent series ${}_r{\phi }_0(0,0,\dots ,0;-;q,x)$.

arXiv admin note: substantial text overlap with arXiv:1203.3404

References in corpus (3)

• Hypergeometric solutions to the q-Painlevé equation of type $A_4^{(1)}$
• Algebraic construction of the Stokes sheaf for irregular linear q-difference equations
• A unified approach to $q$-special functions of the Laplace type