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Mathematics > Classical Analysis and ODEs

arXiv:1806.05375 (math)
[Submitted on 14 Jun 2018 (v1), last revised 5 Mar 2019 (this version, v2)]

Title:The $q$-Borel Sum of Divergent Basic Hypergeometric Series ${}_rφ_s(a;b;q,x)$

Authors: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.
Subjects: Classical Analysis and ODEs (math.CA)
MSC classes: 33D15, 39A13, 34M30
Cite as: arXiv:1806.05375 [math.CA]
  (or arXiv:1806.05375v2 [math.CA] for this version)
  https://doi.org/10.48550/arXiv.1806.05375
Journal reference: SIGMA 15 (2019), 016, 12 pages
Related DOI: https://doi.org/10.3842/SIGMA.2019.016

Submission history

From: Shunya Adachi [view email]
[v1] Thu, 14 Jun 2018 06:02:21 UTC (9 KB)
[v2] Tue, 5 Mar 2019 13:41:23 UTC (14 KB)
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