Modular differential equations and orthogonal polynomials
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916899586048000 |
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| author | Besrour, Khalil Saber, Hicham Sebbar, Abdellah |
| author_facet | Besrour, Khalil Saber, Hicham Sebbar, Abdellah |
| contents | We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the $J$-invariant, reducing the problem to an algebraic system. We show that the roots of this system are captured by orthogonal polynomials satisfying a Fuchsian differential equation. Their recurrence, norms, and weight function are derived, completing the classification of equivariant solutions in this setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10788 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modular differential equations and orthogonal polynomials Besrour, Khalil Saber, Hicham Sebbar, Abdellah Number Theory Classical Analysis and ODEs We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the $J$-invariant, reducing the problem to an algebraic system. We show that the roots of this system are captured by orthogonal polynomials satisfying a Fuchsian differential equation. Their recurrence, norms, and weight function are derived, completing the classification of equivariant solutions in this setting. |
| title | Modular differential equations and orthogonal polynomials |
| topic | Number Theory Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2508.10788 |