A note on Clebsch–Gordan integral, Fourier–Legendre expansions and closed form for hypergeometric series

Abstract

In this paper, we show that a closed-form formula for the generalized Clebsch-Gordan integral and the Fourier-Legendre expansion theory allow to evaluate hypergeometric series involving powers of the normalized central binomial coefficient 1 4 n 2n n . Keywords Hypergeometric functions • Fourier-Legendre expansion • Clebsch-Gordan integral • Complete elliptic integral of the first kind • Closed form

Applications of Caputo operators in the evaluation of Clebsch–Gordan-type multiple elliptic integrals

“…in Theorem 2. For the α = − 16 9 case, we may mimic our above proof to prove (10). For the α = −8 case, we may again mimic our above proof to prove (11).…”

“…As indicated above, Zhou's 2014 article [28] is the main inspiration for our current work. This current work is also inspired by many references citing or otherwise related to Zhou's article [28], including [2,9,10,15,21,24,25,26,27,29,30,31,32], and these references include a number of articles involving definite integrals from 0 to 1 with integrands containing twofold products of complete elliptic expressions [2,9,10,15,27,29] or threefold such products [21,24,30]. These references add to our interest in the multiple elliptic integrals highlighted in Section 1.1 and proved in the current Section.…”

“…for µ and ν in C, and this terminology is also used in the article [10] relevant to much of our work.…”

New Clebsch–Gordan‐type integrals involving threefold products of complete elliptic integrals

“…As indicated above, Zhou's 2014 article [5] is the main inspiration for our current work. This current work is also inspired by many references citing or otherwise related to Zhou's article [5], including previous studies [4,11,[13][14][15][16][17][18][19][20][21][22][23], and these references include a number of articles involving definite integrals from 0 to 1 with integrands containing twofold products of complete elliptic expressions [11, 13-15, 19, 20] or threefold such products [4,16,21]. These references add to our interest in the multiple elliptic integrals highlighted in Section 1.1 and proved in the current section.…”

“…for 𝜇 and 𝜈 in C, and this terminology is also used in the article [11] relevant to much of our work.…”