Some Additions to a Family of Sums and Integrals related to Hurwitz' Zeta Function(s), Euler Polynomials and Euler Numbers.

Abstract

Integrals involving the kernel function sech\((\pi x)\) over a semi-in nite
range are of general interest in the study of Riemann's function \(\zeta(s)\) and Hurwitz'
function \(\zeta(s,a)\). The emphasis here is to evaluate such integrals that include
monomial moments in the integrand. These are evaluated in terms of both \(\zeta(s,a)\) . 
and its alternating equivalent \(\eta(s,a)\), thereby adding some new members to a
known family of related integrals. Additionally, some nite series involving both
Euler polynomials and numbers are summed in closed form. Special cases of these
results are speci cally evaluated and used to verify a claimed connection with the
function \(\zeta\) (2m + 1) for m = 1 and m = 2.