Asymptotic expansions of double gamma-functions and related remarks

Let Gamma(2) (beta, (w1, w2)) be the double gamma-function. We prove asymptotic expansions of log Gamma(2) (beta, (1, w)) with respect to w, both when \w\ --> + infinity and when \w\ 0. Our proof is based on the results on Barnes' double zeta-functions given in the author's former article [12]. We also prove asymptotic expansions of log Gamma(2) (2epsilon(n) - 1, (epsilon(n) - 1, epsilon(n))), log rho(2) (epsilon(n) - and log rho(2)(epsilon(n), epsilon(n)(2) - epsilon(n)), where epsilon(n) is the fundamental unit of K = Q(root4n(2) + 8n + 3). Combining those results with Fujii's formula [6] [7], we obtain an expansion formula for zeta'(1;upsilon(1)), where zeta(s; upsilon(1)) is Hecke's zeta-function associated with K.