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

Mark van Hoeij, Duco van Straten, Wadim Zudilin

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…