A completely monotone function related to the Gamma function

[摘要] We show that the reciprocal of the function f(z) = log Gamma (z + 1)/z log z, z is an element of C \ ] -infinity ,0] is a Stieltjes transform. As a corollary we obtain that the derivative of f is completely monotone, in the sense that (-1)(n-1) f((n)) (x)greater than or equal to1 for all n greater than or equal to 1 and all x > 0. This answers a question raised by Dimitar Dimitrov at the Fifth International Symposium on Orthogonal Polynomials, Special Functions and Applications held in Patras in September 1999, To prove the result we examine the imaginary part of 1/f in the upper half-plane, in particular close to the negative real axis, where Stirling's formula is not valid. (C) 2001 Elsevier Science B.V. All rights reserved.

[发布日期] 2001-08-01

[效力级别]  Proceedings Paper

[关键词] gamma function;completely monotone function