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arXiv · 1806.05375

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

Shunya Adachi
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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.

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BibTeXRIS

Shunya Adachi. 2018-06-14. The $q$-Borel Sum of Divergent Basic Hypergeometric Series ${}_r\varphi_s(a;b;q,x)$. https://doi.org/10.3842/sigma.2019.016

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