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arXiv · 1404.2541

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

Takeshi Morita
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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 ${}_2φ_0(0,0;-;q,x)$. 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 ${}_rφ_0(0,0,\dots ,0;-;q,x)$.

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Takeshi Morita. 2014-04-09. The Stokes phenomenon for the Ramanujan's $q$-difference equation and its higher order extension. https://arxiv.org/abs/1404.2541

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