Let p, q ∈ ℝ with p − q ≽ 0, \(\sigma = {1 \over 2}(p + q - 1)\) and \(s = {1 \over 2}(1 - p + q)\) , and let \({{\cal D}_m}(x;p,q) = {{\cal D}_0}(x;p,q) + \sum\limits_{k = 1}^m {{{{B_{2k}}(s)} \over {2k{{(x + \sigma )}^{2k}}}},} \) where \({{\cal D}_0}(x;p,q) = {{\psi (x + p) + \psi (x + q)} \over 2} - \ln (x + \sigma ).\)
We establish the asymptotic expansion \({{\cal D}_0}(x;p,q) \sim - \sum\limits_{n = 1}^\infty {{{{B_{2n}}(s)} \over {2n{{(x + \sigma )}^{2n}}}}\,\,\,\,\,{\rm{as}}\,\,x \to \infty ,} \) where B2n(s) stands for the Bernoulli polynomials. Further, we prove that the functions \({( - 1)^m}{{\cal D}_m}(x;p,q)\) and \({( - 1)^{m + 1}}{{\cal D}_m}(x;p,q)\) are completely monotonic in x on (−σ, ∞) for every m ∈ ℕ0 if and only if \(p - q \in [0,{1 \over 2}]\) and p − q = 1, respectively. This not only unifies the two known results but also yields some new results.
