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

TL;DR

This paper explores the relationships among hypergeometric functions, elliptic integrals, and Fourier-Legendre series, deriving new transformations and closed-form series evaluations involving constants like Catalan's constant and zeta values.

Contribution

It introduces novel hypergeometric transformations and evaluates series with harmonic numbers using a dual expansion method involving elliptic integrals and FL series.

Findings

Derived new hypergeometric transformations.

Evaluated series involving harmonic numbers and special constants.

Established connections between elliptic integrals and series expansions.

Abstract

Motivated by our previous work on hypergeometric functions and the parbelos constant, we perform a deeper investigation on the interplay among generalized complete elliptic integrals, Fourier-Legendre (FL) series expansions, and ${}_p F_q$ series. We produce new hypergeometric transformations and closed-form evaluations for new series involving harmonic numbers, through the use of the integration method outlined as follows: Letting $K$ denote the complete elliptic integral of the first kind, for a suitable function $g$ we evaluate integrals such as $$ \int_{0}^{1} K\left( \sqrt{x} \right) g(x) \, dx $$ in two different ways: (1) by expanding $K$ as a Maclaurin series, perhaps after a transformation or a change of variable, and then integrating term-by-term; and (2) by expanding $g$ as a shifted FL series, and then integrating term-by-term. Equating the expressions produced by these two approaches often gives us new closed-form evaluations, as in the formulas involving Catalan's constant $G$ $$ \sum _{n = 0}^{\infty } \binom{2 n}{n}^2 \frac{H_{n + \frac{1}{4}} - H_{n-\frac{1}{4}}}{16^{n} } = \frac{\Gamma^4 \left(\frac{1}{4}\right)}{8 \pi^2}-\frac{4 G}{\pi},$$ $$ \sum _{m, n \geq 0} \frac{\binom{2 m}{m}^2 \binom{2 n}{n}^2 }{ 16^{m + n} (m+n+1) (2 m+3) } = \frac{7 \zeta (3) - 4 G}{\pi^2}.$$