The generalized Zwegers' $μ$-function and transformation formulas for the bilateral basic hypergeometric series
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866916125434970112 |
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| author | Shibukawa, Genki Tsuchimi, Satoshi |
| author_facet | Shibukawa, Genki Tsuchimi, Satoshi |
| contents | By applying Slater's transformation formulas for the bilateral basic hypergeometric series ${}_2ψ_{2}$, we derive three type translation formulas for the generalized Zwegers' $μ$-function (``continuous $q$-Hermite function'') which was introduced by Shibukawa--Tsuchimi (SIGMA, 2023). From some Bailey's transformation formula of ${}_2ψ_{2}$, we also give a formula for the expression of the generalized Zwegers' $μ$-function by some very-well-poised bilateral basic hypergeometric series ${}_4ψ_{8}$. As an application of this new expression formula for the generalized Zwegers' $μ$-function, we obtain some new Fourier expansions for the Weierstrass and Jacobi elliptic functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_08101 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The generalized Zwegers' $μ$-function and transformation formulas for the bilateral basic hypergeometric series Shibukawa, Genki Tsuchimi, Satoshi Classical Analysis and ODEs 33D15, 39A13, 33D90, 30D05, 11F50 By applying Slater's transformation formulas for the bilateral basic hypergeometric series ${}_2ψ_{2}$, we derive three type translation formulas for the generalized Zwegers' $μ$-function (``continuous $q$-Hermite function'') which was introduced by Shibukawa--Tsuchimi (SIGMA, 2023). From some Bailey's transformation formula of ${}_2ψ_{2}$, we also give a formula for the expression of the generalized Zwegers' $μ$-function by some very-well-poised bilateral basic hypergeometric series ${}_4ψ_{8}$. As an application of this new expression formula for the generalized Zwegers' $μ$-function, we obtain some new Fourier expansions for the Weierstrass and Jacobi elliptic functions. |
| title | The generalized Zwegers' $μ$-function and transformation formulas for the bilateral basic hypergeometric series |
| topic | Classical Analysis and ODEs 33D15, 39A13, 33D90, 30D05, 11F50 |
| url | https://arxiv.org/abs/2305.08101 |