Complete monotonicity involving the divided difference of polygamma functions

For r, s ? R and ? = min {r, s}, let D[x + r, x + s; ?n?1] ? ??n (x) be the divided difference of the functions ?n?1 = (?1)n ?(n?1) (n ? N) on (??,?), where ?(n) stands for the polygamma functions. In this paper, we present the necessary and sufficient conditions for the functions x ? ?k i=1 ?mi (x) ? ?k ?k i=1 ?ni (x) , x ? ?k i=1 ?ni (x) ? ?k?snk (x) to be completely monotonic on (??,?), where mi, ni ? N for i = 1,..., k with k ? 2 and snk = ?k i=1 ni. These generalize known results and gives an answer to a problem.