Hypergeometric solutions to Schwarzian equations
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866912075725406208 |
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| author | Besrour, Khalil Sebbar, Abdellah |
| author_facet | Besrour, Khalil Sebbar, Abdellah |
| contents | In this paper we study the modular differential equation $y''+s\,E_4\, y=0$ where $E_4$ is the weight 4 Eisenstein series and $s=π^2r^2$ with $r=n/m$ being a rational number in reduced form such that $m\geq 7$. This study is carried out by solving the associated Schwarzian equation $\{h,τ\}=2\,s\,E_4$ and using the theory of equivariant functions on the upper half-plane and the 2-dimensional vector-valued modular forms. The solutions are expressed in terms of the Gauss hypergeometric series. This completes the study of the above-mentioned modular differential equation of the associated Schwarzian equation given that the cases $1\leq m\leq 6$ have already been treated in the litterature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04973 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypergeometric solutions to Schwarzian equations Besrour, Khalil Sebbar, Abdellah Number Theory Classical Analysis and ODEs 11F03, 11F11, 34M05 In this paper we study the modular differential equation $y''+s\,E_4\, y=0$ where $E_4$ is the weight 4 Eisenstein series and $s=π^2r^2$ with $r=n/m$ being a rational number in reduced form such that $m\geq 7$. This study is carried out by solving the associated Schwarzian equation $\{h,τ\}=2\,s\,E_4$ and using the theory of equivariant functions on the upper half-plane and the 2-dimensional vector-valued modular forms. The solutions are expressed in terms of the Gauss hypergeometric series. This completes the study of the above-mentioned modular differential equation of the associated Schwarzian equation given that the cases $1\leq m\leq 6$ have already been treated in the litterature. |
| title | Hypergeometric solutions to Schwarzian equations |
| topic | Number Theory Classical Analysis and ODEs 11F03, 11F11, 34M05 |
| url | https://arxiv.org/abs/2403.04973 |