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

TL;DR

This paper develops a $q$-Borel summation method for divergent basic hypergeometric series, revealing a $q$-analog of the Stokes phenomenon and connecting to classical hypergeometric sums as $q$ approaches 1.

Contribution

It introduces a novel $q$-Borel summability approach for divergent basic hypergeometric series, providing explicit solutions and analyzing their asymptotic behavior.

Findings

Established $q$-Borel summability for basic hypergeometric series.

Demonstrated the $q$-analog of the Stokes phenomenon.

Connected the $q$-analogs to classical hypergeometric sums as $q\to1$.

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.