On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier–Legendre 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 Fqp 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∫01K(x)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∑n=0∞(2nn)2Hn+14−Hn−1416n=Γ4(14)8π2−4Gπ,∑m,n≥0(2mm)2(2nn)216m+n(m+n+1)(2m+3)=7ζ(3)−4Gπ2.