On Ruijsenaars’ asymptotic expansion of the logarithm of the double gamma function

We study the remainder RN(x) in an asymptotic expansion due to S.N.M. Ruijsenaars, for the logarithm of the double gamma function. We show that for any even number N the function (−1) N2 −1RN(x) is completely monotonic on (0,∞). This proves a recent conjecture of H.L. Pedersen. In addition, we give an estimate for the remainder RN(x) for N even. © 2007 Elsevier Inc. All rights reserved