On a higher-order version of a formula due to Ramanujan

Abstract

A special case of an Entry in Part II of Ramanujan's Notebooks is such that 1 + 1 5 ( 1 2 ) 2 + 1 9 ( 1 ⋅ 3 2 ⋅ 4 ) 2 + ⋯ = Γ 4 ( 1 4 ) 16 π 2 . This formula leads us to consider the higher-order version of the above series given by replacing the squares of normalized central binomial coefficients with fourth powers. Using a Fourier–Legendre expansion introduced in a 2022 article by Cantarini, together with a multiple elliptic integral evaluation conjectured by Wan and proved by Zhou, we prove the very natural extension shown below of Ramanujan's formula: 1 + 1 5 ( 1 2 ) 4 + 1 9 ( 1 ⋅ 3 2 ⋅ 4 ) 4 + ⋯ = Γ 8 ( 1 4 ) 96 π 5 . Furthermore, and in a closely related way, we show how a main result in an article by Papanikolas et al. concerning a Calabi–Yau threefold is equivalent to the evaluation of a Clebsch–Gordan-type multiple elliptic integral related to the work of Zhou and Brychkov.