Eulerian fractions and Stirling, Bernoulli and Euler functions with complex order parameters and their impact on the polylogarithm function

Abstract

We first study some generalizations of Eulerian fractions with complex order parameter and investigate their interrelationship with likewise generalized Eulerian functions as well as Stirling functions. We apply the new approach to polylogarithms of non-integral order, for which only a few values are known in closed form. In particular, we present a structural solution of the counterpart of an old conjecture of Mengoli and Euler in the polylogarithm case with the aid of Riemann?s zeta function and the Dirichlet eta and beta functions.