Stieltjes continued fractions for polygamma functions; speed of convergence
For \(k\geq 0\), the polygamma function \(\Psi_{k}(z)\) is defined by \[ \Psi_{k}(z):=\frac{d^{k+1}}{dz^{k+1}}\operatorname{Log} \Gamma(z),\quad z\in \mathbb{C} \setminus \{0,-1,-2,\ldots \}. \] These functions have a Stieltjes continued fraction representation of the form \[ \Psi_{0}(z)=\operatorname{Log}\,z-\frac{1}{2z}-g(1,\,z) \] and \[ \Psi_{k}(z)=\frac{(-1)^{k-1}}{z^{k}}\biggl[ (k-1)!+\frac{k!}{2z}+(2\pi)^{k}g(k+1,\,z)\biggr], \quad k\geq 1, \] where \[ g(k,z)= \frac{a_{1}^{(k)}}{z^2}_{+} \frac{a_{2}^{(k)}}{1}_{+} \frac{a_{3}^{(k)}}{z^2}_{+}\frac{a_{4}^{(k)}}{1}_{+\cdots}, \quad a_{m}^{(k)}>0 \] the convergence being locally uniform on \(| \arg z| <\pi/2\). The authors show that for each \(k=0,1,2,\ldots\), the coefficients \(a_{m}^{(k)}\) in the expansion above satisfy the asymptotic property \[ a_{m}^{(k)} \sim \frac{m^2}{16}, \quad m \rightarrow \infty, \] and then that for each \(k \geq 0\) and each \(z\in \mathbb{C}\) with \(| \arg z| <\pi/2\), there exist constants \(A>0\) and \(B>0\) such that \[ | g(k,\,z)-g_{n}(k,z)| \leq \frac{A}{n^{B}},\quad n\geq 1, \] where \(g_{n}(k,z)\) denotes the \(n\)th approximant of the Stieltjes continued fraction \(g(k,z)\).
scientific article; zbMATH DE number 2173188