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A generating function of the squares of Legendre polynomials. (English) Zbl 1334.33022
This note relates a one-parametric generating function for the square of the Legendre polynomials \(P_n (y)\) to an arithmetic hypergeometric series parametrised by a level 7 modular function in the form \[ \sum^\infty_{n=0}\binom{2n}{n}P^2_n (y)z^n = \frac{1 +2v}{1 + 4v}\sum^\infty_{n=0} u_n\Big(\frac{v}{(1 + 4v)^3}\Big)^n, \] where \[ y=\sqrt{\frac{(1 + v)(1 + 8v)}{1 + 5v + 8v^2}}, \quad z= \frac{v(1 + 5v + 8v^2 )}{(1 + 2v)^2 (1 + 4v)^2} \] and \(u_n\) is defined as a sum involving \(n + 1\) terms which satisfies a three-term Apéry-like recurrence relation.
Use of the above modular parametrisation enables a subfamily of identities for \(1/\pi\) to be identified. These identities had been experimentally observed by Sun.

MSC:
33C20 Generalized hypergeometric series, \({}_pF_q\)
11F03 Modular and automorphic functions
11F11 Holomorphic modular forms of integral weight
11Y60 Evaluation of number-theoretic constants
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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