# Convergence of ray sequences of Pade approximants to 2F1(a,1;c;z), c\>a\>0 Source: [Convergence of ray sequences of Pade approximants to 2F1(a,1;c;z), c\>a\>0]() Collection index: [llms.txt]() Data updated: 2025-10-28T13:55:10.627Z Authors: K Driver, K Jordaan Published: 2009-01-05T08:49:35.000Z ## Abstract The Pad\\'e table of $\\phantom{}\_2F\_1(a,1;c;z)$ is normal for $c\>a\>0$ (cf. \\cite{3}). For $m \\geq n-1$ and $c \\notin {\\zz}^{\\phantom{}^-}$, the denominator polynomial $Q\_{mn}(z)$ in the $\[m/n\]$ Pad\\'e approximant $P\_{mn}(z)/Q\_{mn}(z)$ for $\\phantom{}\_2F\_1(a,1;c;z)$ and the remainder term $Q\_{mn}(z)\\phantom{}\_2F\_1(a,1;c;z)-P\_{mn}(z)$ were explicitly evaluated by Pad\\'e (cf. \\cite{2}, \\cite{5} or \\cite{7}). We show that for $c\>a\>0$ and $m\\geq n-1$, 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 $\\phantom{}\_2F\_1(a,1;c;z)$ as $m \\to \\infty$, $ n/m \\to \\rho$ with $0\<\\rho \\leq 1$, uniformly on compact subsets of the unit disc $\|z\|\<1$ for $c\>a\>0$ ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()