The multiple Hurwitz zeta function and the multiple Hurwitz-Euler ETA function

摘要

Almost eleven decades ago, Barnes introduced and made a systematic investigation on the multiple Gamma functions Γ_n. 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 ζ _n(s, a). In this paper, we first aim at presenting a generalized Hurwitz formula for ζ _n (s, a) together with its various special cases. Secondly, we give analytic continuations of multiple Hurwitz-Euler eta function η_n (s, a) in two different ways. As a by-product of our second investigation, a relationship between η_n(-l,a) (l ∈ N_0) and the generalized Euler polynomials E ~n_l(n-a) is also presented.