부산대학교 도서관

A special case of an Entry in Part II of Ramanujan's Notebooks is such that \[ 1+\frac{1}{5} \left(\frac{1}{2} \right)^{2} + \frac{1}{9} \left(\frac{1\cdot 3}{2 \cdot 4} \right)^{2} + \cdots = \frac{\Gamma^4 \left(\frac{1}{4}\right)}{16\pi ^2}. \] 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: \[ 1 + \frac{1}{5} \left(\frac{1}{2} \right)^{4} + \frac{1}{9} \left(\frac{1\cdot 3}{2\cdot 4} \right)^{4}+\cdots =\frac{\Gamma^8 \left(\frac{1}{4}\right)}{96\pi ^5}. \] 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. [ABSTRACT FROM AUTHOR]