ASYMPTOTIC EXPANSIONS AND COMPLETE MONOTONICITY PROPERTIES FOR THE BARNES G-FUNCTION

Abstract

We present several new asymptotic expansions of the logarithm of the Barnes $G$-function in terms of the gamma function and the polygamma functions. We also give several explicit formulas and recurrence relations for determining the coefficients in these asymptotic expansions. By using the results obtained, we derive recursion formulas for the Bernoulli numbers and the corresponding results for the Riemann zeta function. Finally, based on the asymptotic expansions which we have presented here, we prove new complete monotonicity properties of some functions involving the Barnes $G$-function, the gamma function and the polygamma functions.