The zeros of Euler's Psi function and its derivatives (Q882042)

The aim of present paper is to study Euler's Psi-function \[ \psi(x):=-\frac{1}{x}-\gamma-\sum_{n=1}^{\infty}\left(\frac{1}{x+n}-\frac{1}{n}\right), \] with \(\gamma=0577\cdots\) Euler's constant, to investigate the locations of the points of inflexion and the positions of the stationary points of its derivative. The author works with the functions \(F_1(x)=\psi(-x)+\gamma\) and \[ F_k(x)=\sum_{n=0}^{\infty}\frac{1}{(x-n)^k}=\frac{-1}{k-1}F'_{k-1}(x)=\frac{(-1)^{k-1}}{(k-1)!}F_1^{(k-1)}(x). \] He gives some trigonometric approximations for the functions \(F_1(x)\) and \(F_2(x)\), which end to some bounds for the zeros of \(\psi\). Finally, he investigates the properties of the horizontal distance between successive branches of the graph of \(F_1\) and consequently \(\psi\).