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

Abstract

We decompose the generating function ∑ (2n choose n) P_n(y)² zⁿ of the squares of Legendre polynomials as a product of periods of hyperelliptic curves. These periods satisfy a family of second order differential equations. This is highly unusual, since four is the expected order for genus 2. These second order equations are arithmetic and yet, surprisingly, their monodromy group is dense in SL₂(R). This suggests that they cannot be solved in terms of hypergeometric functions, which is novel for arithmetic second order differential equations that are defined over Q, and also novel for a family of such equations. We complement our analysis with a recipe for constructing similar examples.