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

TL;DR

This paper proves the convergence of certain Padé approximants to the hypergeometric function _2F_1(a,1;c;z) for c>a>0, showing that the poles lie on the cut and the approximants converge uniformly inside the unit disk.

Contribution

It explicitly evaluates the denominator polynomials and remainder terms of Padé approximants for _2F_1(a,1;c;z) and proves their convergence and pole distribution.

Findings

Poles of Padé approximants lie on the cut (1,∞).

Sequence of approximants converges uniformly on compact subsets of |z|<1.

Explicit expressions for denominator polynomials and remainders are provided.

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 $c>a>0$