On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier–Legendre expansions a, b c John M. Campbell ∗, Jacopo D’Aurizio , Jonathan Sondow aDepartment of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3 bDepartment of Mathematics, University of Pisa, Italy, Largo Bruno Pontecorvo 5 - 56127 c209 West 97th Street, New York, NY 10025 Abstract Motivated by our previous work on hypergeometric functions and the parbelos constant, we perform a deeper investigation on the interplay among general- ized complete elliptic integrals, Fourier–Legendre (FL) series expansions, and pFq 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 1 K √x g(x) dx Z0 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 arXiv:1710.03221v4 [math.NT] 13 Feb 2019 closed-form evaluations, as in the formulas involving Catalan’s constant G 2 4 1 ∞ 2n Hn+ 1 Hn 1 Γ 4G 4 − − 4 = 4 , n 16n 8π2 − π n=0 X 2 2 2m 2n 7ζ(3) 4G m n = − . 16m+n(m + n + 1)(2m + 3) π2 m,n 0 X≥ ∗Corresponding author Email address:
[email protected] (John M.