A NEW PROPERTY OF THE WALLIS POWER FUNCTION

The Wallis power function is defined on (- min {p, q} , oo) by Gamma (x +p) 1/(p-q) Wp,q (x) = if p =?q and Wp,p (x) = e psi (x+p). Gamma (x + q) Let pi, qi E R with 0 < <delta>i = pi - qi 1, theta i = (1 - delta i) /2 for i = 1,2 and p1 + q1 = p2 + q2 = 2 sigma + 1. We prove that, if q1 q2 then for any integer m E N, the function [?m x ?-> (-1)m ln Wp1,q1 (x) a2k (theta 1, theta 2) Wp2,q2 (x) - (2k + 1) (2k) (x + sigma )2k k=1 is completely monotonic on (-sigma, oo), where B2k+1 (theta 2) B2k+1 (theta 1) a2k (theta 1, theta 2) = - . theta 2 - 1/2 theta 1 - 1/2 This extends and generalizes some known results.