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

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Abstract: We decompose the generating function

of the squares of Legendre polynomials as a product of periods of hyperelliptic curves. These periods satisfy a family of

e x t i t s e c o n d

order differential equations. This is highly unusual since

e x t i t f o u r

is the expected order for genus 2. These second order equations are arithmetic and yet, surprisingly, their monodromy group is dense in

o p e r a t o r n a m e {S L}_{2} (m a t h b b R)

. This implies that they cannot be solved in terms of hypergeometric functions, which is novel for arithmetic second order differential equations that are

e x t i t d e f i n e d o v e r

m a t h b b Q

, and also novel for a

e x t i t f a m i l y

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/.

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