Convergence of ray sequences of Pade approximants to 2F1(a,1;c;z), c>a>0

Abstract

The Padé table of ${}_2{F}_1(a,1;c;z)$ is normal for $c>a>0$ (cf. \cite{3}). For $m\ge n-1$ and $c \notin {\zz}^{\phantom{}^-}$, the denominator polynomial $Q_{mn}(z)$ in the $[m/n]$ Padé approximant $P_{mn}(z)/Q_{mn}(z)$ for ${}_2{F}_1(a,1;c;z)$ and the remainder term $Q_{mn}(z){}_2{F}_1(a,1;c;z)-P_{mn}(z)$ were explicitly evaluated by Padé (cf. \cite{2}, \cite{5} or \cite{7}). We show that for $c>a>0$ and $m\ge n-1$, the poles of $P_{mn}(z)/Q_{mn}(z)$ lie on the cut $(1,\infty )$. We deduce that the sequence of approximants $P_{mn}(z)/Q_{mn}(z)$ converges to ${}_2{F}_1(a,1;c;z)$ as $m\to \infty$, $n/m\to \rho$ with $0<\rho \le 1$, uniformly on compact subsets of the unit disc $|z|<1$ for $c>a>0$