The Multiple Hurwitz Zeta Function and the Multiple Hurwitz-Euler Eta Function

Almost eleven decades ago, Barnes introduced and made a systematic investigation on the multiple Gamma functions Γₙ. In about the middle of 1980s, these multiple Gamma functions were revived in the study of the determinants of Laplacians on the n-dimensional unit sphere Sⁿ by using the multiple Hurwitz zeta functions ζₙ(s,a). In this paper, we first aim at presenting a generalized Hurwitz formula for ζₙ(s,a) together with its various special cases. Secondly, we give analytic continuations of multiple Hurwitz-Euler eta function ηₙ(s,a) in two different ways. As a by-product of our second investigation, a relationship between ηₙ(-,a) ( ∈ N₀) and the generalized Euler polynomials E_⁽ⁿ⁾(n-a) is also presented.