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

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 F-p(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 integral(1)(0) K (root 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 Sigma(infinity)(n=0) ((2n)(n))(2) Hn+1/4 - Hn-1/4/16(n) = Gamma(4)(1/4)/8 pi(2) - 4G/pi, Sigma(m, n) (>= )(0) ((m) (2m))(2) ((n) (2n))(2)/16(m+n)(m+n+1)(2m+3) = 7 zeta(3) - 4(G)/pi(2). (C) 2019 Elsevier Inc. All rights reserved.