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

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