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

TL;DR

This paper presents a closed-form formula for the generalized Clebsch-Gordan integral and uses Fourier-Legendre expansions to evaluate hypergeometric series involving binomial coefficients.

Contribution

It introduces a novel closed-form expression for the Clebsch-Gordan integral and applies Fourier-Legendre expansions to evaluate complex hypergeometric series.

Findings

Closed-form formula for the generalized Clebsch-Gordan integral.

Method to evaluate hypergeometric series with binomial coefficients.

Application of Fourier-Legendre expansions in series evaluation.

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 ${\displaystyle \frac{1}{4^{n}}\dbinom{2n}{n}}$.