Some properties of the divided difference of psi and polygamma functions

Let ψn=(−1)n−1ψ(n) for n≥0, where ψ(n) stands for the psi and polygamma functions. For p,q∈R and ρ=min⁡(p,q), letD[x+p,x+q;ψn−1]≡−ϕn(x) be the divided difference of the functions ψn−1 for x>−ρ. In this paper, we establish the necessary and sufficient conditions for the functionΦn(x,λ)=ϕn+1(x)2−λϕn(x)ϕn+2(x) to be completely monotonic on (−ρ,∞). In particular, we find that the function ψn+12/(ψnψn+2) is strictly decreasing from (0,∞) onto (n/(n+1),(n+1)/(n+2)). These not only generalize and strengthen some known results, but also yield many new and interesting ones.