# zbMATH — the first resource for mathematics

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.)
Full Text:
##### References:
 [1] DOI: 10.1017/S0305004112000254 · Zbl 1268.11165 [2] DOI: 10.1016/j.jnt.2007.08.015 · Zbl 1218.11043 [3] DOI: 10.1090/S0002-9939-1951-0045875-2 [4] DOI: 10.1112/jlms/s1-13.1.8 · Zbl 0018.12202 [5] DOI: 10.1016/j.aim.2011.06.011 · Zbl 1234.33009 [6] DOI: 10.1016/j.jat.2011.12.001 · Zbl 1242.33018 [7] Maximon, Norske Vid. Selsk. Forh. Trondheim 29 pp 82– (1956) [8] DOI: 10.1007/s11139-011-9357-3 · Zbl 1336.11031 [9] DOI: 10.1112/S0025579309000436 · Zbl 1275.11035 [10] Bailey, Generalized Hypergeometric Series (1935) · Zbl 0011.02303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.