The q-Borel Sum of Divergent Basic Hypergeometric Series ᵣφₛ(a; b; q, x)
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...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2019 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2019
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210059 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | The q-Borel Sum of Divergent Basic Hypergeometric Series ᵣφₛ(a; b; q, x) / S. Adachi // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 14 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862734550561456128 |
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| author | Adachi, S. |
| author_facet | Adachi, S. |
| citation_txt | The q-Borel Sum of Divergent Basic Hypergeometric Series ᵣφₛ(a; b; q, x) / S. Adachi // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 14 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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→1 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.
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| first_indexed | 2025-12-07T21:24:14Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-210059 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:24:14Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Adachi, S. 2025-12-02T09:31:56Z 2019 The q-Borel Sum of Divergent Basic Hypergeometric Series ᵣφₛ(a; b; q, x) / S. Adachi // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 14 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33D15; 39A13; 34M30 arXiv: 1806.05375 https://nasplib.isofts.kiev.ua/handle/123456789/210059 https://doi.org/10.3842/SIGMA.2019.016 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→1 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. The author would like to thank the referees for their helpful suggestions and valuable comments. Additionally, the author is grateful that one of the referees let him know the existence of the papers [5] and [11]. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The q-Borel Sum of Divergent Basic Hypergeometric Series ᵣφₛ(a; b; q, x) Article published earlier |
| spellingShingle | The q-Borel Sum of Divergent Basic Hypergeometric Series ᵣφₛ(a; b; q, x) Adachi, S. |
| title | The q-Borel Sum of Divergent Basic Hypergeometric Series ᵣφₛ(a; b; q, x) |
| title_full | The q-Borel Sum of Divergent Basic Hypergeometric Series ᵣφₛ(a; b; q, x) |
| title_fullStr | The q-Borel Sum of Divergent Basic Hypergeometric Series ᵣφₛ(a; b; q, x) |
| title_full_unstemmed | The q-Borel Sum of Divergent Basic Hypergeometric Series ᵣφₛ(a; b; q, x) |
| title_short | The q-Borel Sum of Divergent Basic Hypergeometric Series ᵣφₛ(a; b; q, x) |
| title_sort | q-borel sum of divergent basic hypergeometric series ᵣφₛ(a; b; q, x) |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210059 |
| work_keys_str_mv | AT adachis theqborelsumofdivergentbasichypergeometricseriesrφsabqx AT adachis qborelsumofdivergentbasichypergeometricseriesrφsabqx |