The $q$-Borel Sum of Divergent Basic Hypergeometric Series ${}_r\varphi_s(a;b;q,x)$
Classical Analysis and ODEs
2019-03-06 v2
Abstract
We study the divergent basic hypergeometric series which is a -analog of divergent hypergeometric series. This series formally satisfies the linear -difference equation. In this paper, for that equation, we give an actual solution which admits basic hypergeometric series as a -Gevrey asymptotic expansion. Such an actual solution is obtained by using -Borel summability, which is a -analog of Borel summability. Our result shows a -analog of the Stokes phenomenon. Additionally, we show that letting 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.
Keywords
Cite
@article{arxiv.1806.05375,
title = {The $q$-Borel Sum of Divergent Basic Hypergeometric Series ${}_r\varphi_s(a;b;q,x)$},
author = {Shunya Adachi},
journal= {arXiv preprint arXiv:1806.05375},
year = {2019}
}