A hyperelliptic saga on a generating function of the squares of Legendre polynomials

Abstract

We decompose the generating function ${\sum }_{n=0}^{\infty }\left(\genfrac{}{}{0ex}{}{2n}{n}\right)P_n(y{)}^2{z}^n$ of the squares of Legendre polynomials as a product of periods of hyperelliptic curves. These periods satisfy a family of $\text{second}$ order differential equations. This is highly unusual since $\text{four}$ is the expected order for genus 2. These second order equations are arithmetic and yet, surprisingly, their monodromy group is dense in ${\mathrm{SL}}_2(\mathbb{R})$. This suggests that they cannot be solved in terms of hypergeometric functions, which is novel for arithmetic second order differential equations that are $\text{defined over}$ $\mathbb{Q}$, and also novel for a $\text{family}$ 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/.

$3^3$ pages, $2^3$ figures; v4: final version accepted for publication