Complete monotonicity of the remainder in an asymptotic series related to the psi function

Let p, q is an element of R with p - q >= 0, sigma = 1/2 (p + q - 1) and s = 1/2 (1 - p + q), and let D-m(x; p, q) = D-0(x; p, q) + Sigma(m)(k=1) B-2k(s)/2k (x + sigma)(2k) where D0(x;p,q) = psi(x+p) + psi(x+q)/2 - ln(x+sigma). We establish the asymptotic expansion D-0(x; p, q) similar to - Sigma(infinity)(n=1) B-2n(s)/2n(x + sigma)(2n) as x -> infinity, where B-2n (s) stands for the Bernoulli polynomials. Further, we prove that the functions (-1)D-m(m) (x; p, q) and (-1)D-m+1(m)(x; p, q) are completely monotonic in x on (-sigma, infinity) for every m is an element of N-0 if and only if p - q is an element of [0, 1/2] and p - q = 1, respectively. This not only unifies the two known results but also yields some new results.