Generating functions of Legendre polynomials: A tribute to Fred Brafman

A new series for π3and related congruences

“…Sun [33,39] 2 , In general, the corresponding p-adic congruences of these seven-type series involve linear combinations of two Legendre symbols. The author's conjectural series of types I-V and VII were studied in [6,48,53]. The author's three conjectural series of type VI and two series of type VII remain open.…”

“…Wan and Zudilin [48] obtained the following irrational series for 1/π involving the Legendre polynomials and the numbers w n : Using our congruence approach (including Conjecture 1.4), we find five rational series for 1/π involving T n (b, c) and the numbers w n ; Theorem 1 of [48] might be helpful to solve them. Remark 9.1.…”

On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier–Legendre expansions

“…New series for 1 π are given in [18] through a generalization of Bailey's identity for 50 generating functions given by componentwise products of Apéry-type sequences and the sequence of Legendre polynomials. The construction of hypergeometric series identities using expansions in terms of Legendre polynomials has practical applications in mathematical physics [14] and related areas; a variety of binomial sum identities given in terms of generalized hypergeometric functions are proved 55 in [11] through the use of the family {P n (x) : n ∈ N 0 }.…”

Short Walk Adventures

“…The example given in Theorem 1 corresponds to the first (orthogonal) situation: on the level of Lie groups, O 2,2 can be realised as the tensor product of two copies of SL 2 (or GL 2 ). There is a limited amount of further examples of this type [21,29,33] though we expect that all underlying Picard-Fuchs differential equations with such monodromy can be represented as tensor products of two arithmetic differential equations of order 2. There is a natural hypergeometric production of such orthogonal cases using Orr-type formulae (see [18,28]) but there are plenty of other cases coming from classical work of W. N. Bailey and its recent generalisations [29,34].…”

“…We remark that, using the general Bailey-Brafman formula and its generalisation from [29], the proof above extends to the factorisation of the two-variable generating functions for an Apéry-like sequence u 0 , u 1 , u 2 , . .…”