### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 016, 12 pages      arXiv:1806.05375      https://doi.org/10.3842/SIGMA.2019.016

### The $q$-Borel Sum of Divergent Basic Hypergeometric Series ${}_r\varphi_s(a;b;q,x)$

Graduate School of Education, Aichi University of Education, Kariya 448-8542, Japan

Received June 15, 2018, in final form February 24, 2019; Published online March 05, 2019

Abstract
We study the divergent basic hypergeometric series which is a $q$-analog of divergent hypergeometric series. This series formally satisfies the linear $q$-difference equation. In this paper, for that equation, we give an actual solution which admits basic hypergeometric series as a $q$-Gevrey asymptotic expansion. Such an actual solution is obtained by using $q$-Borel summability, which is a $q$-analog of Borel summability. Our result shows a $q$-analog of the Stokes phenomenon. Additionally, we show that letting $q\to1$ in our result gives the Borel sum of classical hypergeometric series. The same problem was already considered by Dreyfus, but we note that our result is remarkably different from his one.

Key words: basic hypergeometric series; $q$-difference equation; divergent power series solution; $q$-Borel summability; $q$-Stokes phenomenon.

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