On some families of integrals solvable in terms of polygamma and negapolygamma functions

Abstract

Beginning with Hermite's integral representation of the Hurwitz zeta function, we derive explicit expressions in terms of elementary, polygamma, and negapolygamma functions for several families of integrals of the type $\int_0^\infty f(t)K(q,t)dt$ with kernels $K(q,t)$ equal to $(e^{2\pi q t}-1)^{-1}$, $(e^{2\pi q t}+1)^{-1}$, and $(\sinh(2\pi q t)^{-1}$.