Some properties of the divided difference of psi and polygamma functions.

Let ψ n = ( − 1 ) n − 1 ψ ( n ) for n ≥ 0 , where ψ ( n ) stands for the psi and polygamma functions. For p , q ∈ R and ρ = min ⁡ ( p , q ) , let D [ x + p , x + q ; ψ n − 1 ] ≡ − ϕ n ( x ) be the divided difference of the functions ψ n − 1 for x > − ρ . In this paper, we establish the necessary and sufficient conditions for the function Φ n ( x , λ ) = ϕ n + 1 ( x ) 2 − λ ϕ n ( x ) ϕ n + 2 ( x ) to be completely monotonic on ( − ρ , ∞ ) . In particular, we find that the function ψ n + 1 2 / ( ψ n ψ n + 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. [ABSTRACT FROM AUTHOR]