Harmonic sums from the Kummer theorem

The Kummer summation theorem for F-2(1)(-1)-series is extended with three integer parameters and then examined by means of power series expansions. Quite a variety of new alternating series involving harmonic-like numbers and squared central binomial coefficients are evaluated in closed form, by making use of coefficient extraction methods that have recently been applied in the determination of explicit identities for infinite families of rational series for 1/pi involving harmonic numbers. The same approach is also applied to obtain new identities for integrals involving complete elliptic integrals with complex arguments. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The Kummer summation theorem for F-2(1)(-1)-series is extended with three integer parameters and examined by power series expansions, leading to various new alternating series involving harmonic-like numbers and squared central binomial coefficients. Coefficient extraction methods are used to obtain closed form evaluations for these series, and new identities for integrals involving complete elliptic integrals with complex arguments are also derived using the same approach.