A Generalization of Zwegers' -Function According to the -Hermite-Weber Difference Equation
We introduce a one-parameter deformation of the Zwegers' -function as the image of -Borel and -Laplace transformations of a fundamental solution for the -Hermite-Weber equation. We further give some formulas for our generalized -function, for example, forward and backward shift, translation, sy...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2023 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2023
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211870 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | A Generalization of Zwegers' -Function According to the -Hermite-Weber Difference Equation. Genki Shibukawa and Satoshi Tsuchimi. SIGMA 19 (2023), 014, 23 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862581058588901376 |
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| author | Shibukawa, Genki Tsuchimi, Satoshi |
| author_facet | Shibukawa, Genki Tsuchimi, Satoshi |
| citation_txt | A Generalization of Zwegers' -Function According to the -Hermite-Weber Difference Equation. Genki Shibukawa and Satoshi Tsuchimi. SIGMA 19 (2023), 014, 23 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We introduce a one-parameter deformation of the Zwegers' -function as the image of -Borel and -Laplace transformations of a fundamental solution for the -Hermite-Weber equation. We further give some formulas for our generalized -function, for example, forward and backward shift, translation, symmetry, a difference equation for the new parameter, and bilateral -hypergeometric expressions. From one point of view, the continuous -Hermite polynomials are some special cases of our -function, and the Zwegers' -function is regarded as a continuous -Hermite polynomial of ''−1 degree''.
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| first_indexed | 2026-03-13T17:26:32Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211870 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T17:26:32Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Shibukawa, Genki Tsuchimi, Satoshi 2026-01-14T09:54:09Z 2023 A Generalization of Zwegers' -Function According to the -Hermite-Weber Difference Equation. Genki Shibukawa and Satoshi Tsuchimi. SIGMA 19 (2023), 014, 23 pages 1815-0659 2020 Mathematics Subject Classification: 33D15; 39A13; 30D05; 11F50; 33D70 arXiv:2206.15137 https://nasplib.isofts.kiev.ua/handle/123456789/211870 https://doi.org/10.3842/SIGMA.2023.014 We introduce a one-parameter deformation of the Zwegers' -function as the image of -Borel and -Laplace transformations of a fundamental solution for the -Hermite-Weber equation. We further give some formulas for our generalized -function, for example, forward and backward shift, translation, symmetry, a difference equation for the new parameter, and bilateral -hypergeometric expressions. From one point of view, the continuous -Hermite polynomials are some special cases of our -function, and the Zwegers' -function is regarded as a continuous -Hermite polynomial of ''−1 degree''. We are grateful to Professor Yasuhiko Yamada (Kobe University) for his helpful advice on our paper. We also wish to thank Professor Yosuke Ohyama (Tokushima University) for his valuable suggestions on -special functions, including in the unpublished proof of Lemma 2.2 [17]. We are also indebted to Professor Kazuhiro Hikami (Kyushu University) for his information on [7] and quantum invariants. Professor Toshiki Matsusaka (Kyushu University) also provides information on [7] and his note [15]. Some pieces of information on the -Appell hypergeometric function Φ(1) and its -difference equations are provided by Dr. T. Nobukawa. Finally, we thank the referees for their helpful comments about mock and indefinite theta functions. This work was supported by JSPS KAKENHI Grant Number 21K13808. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Generalization of Zwegers' -Function According to the -Hermite-Weber Difference Equation Article published earlier |
| spellingShingle | A Generalization of Zwegers' -Function According to the -Hermite-Weber Difference Equation Shibukawa, Genki Tsuchimi, Satoshi |
| title | A Generalization of Zwegers' -Function According to the -Hermite-Weber Difference Equation |
| title_full | A Generalization of Zwegers' -Function According to the -Hermite-Weber Difference Equation |
| title_fullStr | A Generalization of Zwegers' -Function According to the -Hermite-Weber Difference Equation |
| title_full_unstemmed | A Generalization of Zwegers' -Function According to the -Hermite-Weber Difference Equation |
| title_short | A Generalization of Zwegers' -Function According to the -Hermite-Weber Difference Equation |
| title_sort | generalization of zwegers' -function according to the -hermite-weber difference equation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211870 |
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