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

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