New properties of the divided difference of psi and polygamma functions

Abstract

Let $\psi _{n}=\left( -1\right) ^{n-1}\psi ^{\left( n\right) }$ for $n\ge 0$ , where $\psi ^{\left( n\right) }$ stands for the psi and polygamma functions. For $p,q\in \mathbb {R}$ and $\rho =\min \left( p,q\right)$, let

$$\begin{aligned} D\left[ x+p,x+q;\psi _{n-1}\right] \equiv -\phi _{n}\left( x\right) \end{aligned}$$

be the divided difference of the functions $\psi _{n-1}$ for $x>-\rho$. In this paper, we prove that for $\left| p-q\right| \gtrless 1$, the sequences $\{\xi _{n}\left( x\right) \}_{n\in \mathbb {N}}$ and $\{\eta _{n}\left( x\right) \}_{n\in \mathbb {N}}$ defined by

$$\begin{aligned} \xi _{n}\left( x\right) =\frac{n\phi _{n}\left( x\right) }{\phi _{n+1}\left( x\right) }-x\text { and }\eta _{n}\left( x\right) =\left[ \frac{\phi _{n}\left( x\right) }{\left( n-1\right) !}\right] ^{-1/n}-x \end{aligned}$$

are strictly decreasing (increasing) and converge to $\min \left( p,q\right)$. Furthermore, if $\left| p-q\right| \gtrless 1$, then the functions $x\mapsto \xi _{n}\left( x\right)$ and $x\mapsto \eta _{n}\left( x\right)$ are strictly increasing (decreasing) from $\left( -\rho ,\infty \right)$ onto $\left( \rho ,\left( p+q-1\right) /2\right)$ ($\left( \left( p+q-1\right) /2,\rho \right)$). These not only generalize and strengthen some known results, but also yield several new sharp bounds involving polygamma functions.