A hyperelliptic saga on a generating function of the squares of Legendre polynomials
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| Format: | Preprint |
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2023
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| _version_ | 1866915471319629824 |
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| author | van Hoeij, Mark van Straten, Duco Zudilin, Wadim |
| author_facet | van Hoeij, Mark van Straten, Duco Zudilin, Wadim |
| contents | We decompose the generating function $\sum_{n=0}^\infty\binom{2n}nP_n(y)^2z^n$ of the squares of Legendre polynomials as a product of periods of hyperelliptic curves. These periods satisfy a family of $\textit{second}$ order differential equations. This is highly unusual since $\textit{four}$ is the expected order for genus 2. These second order equations are arithmetic and yet, surprisingly, their monodromy group is dense in $\operatorname{SL}_2(\mathbb{R})$. This suggests that they cannot be solved in terms of hypergeometric functions, which is novel for arithmetic second order differential equations that are $\textit{defined over}$ $\mathbb{Q}$, and also novel for a $\textit{family}$ of such equations. We complement our analysis with a recipe for constructing similar examples.
Maple's support for the paper is available at https://www.math.fsu.edu/~hoeij/saga/. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_04921 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A hyperelliptic saga on a generating function of the squares of Legendre polynomials van Hoeij, Mark van Straten, Duco Zudilin, Wadim Number Theory Algebraic Geometry Classical Analysis and ODEs Combinatorics 11F99, 11Y60, 14H45, 14Q05, 33C20, 33E30, 34M35 We decompose the generating function $\sum_{n=0}^\infty\binom{2n}nP_n(y)^2z^n$ of the squares of Legendre polynomials as a product of periods of hyperelliptic curves. These periods satisfy a family of $\textit{second}$ order differential equations. This is highly unusual since $\textit{four}$ is the expected order for genus 2. These second order equations are arithmetic and yet, surprisingly, their monodromy group is dense in $\operatorname{SL}_2(\mathbb{R})$. This suggests that they cannot be solved in terms of hypergeometric functions, which is novel for arithmetic second order differential equations that are $\textit{defined over}$ $\mathbb{Q}$, and also novel for a $\textit{family}$ of such equations. We complement our analysis with a recipe for constructing similar examples. Maple's support for the paper is available at https://www.math.fsu.edu/~hoeij/saga/. |
| title | A hyperelliptic saga on a generating function of the squares of Legendre polynomials |
| topic | Number Theory Algebraic Geometry Classical Analysis and ODEs Combinatorics 11F99, 11Y60, 14H45, 14Q05, 33C20, 33E30, 34M35 |
| url | https://arxiv.org/abs/2306.04921 |