The expansion of a finite number of terms of the Gauss hypergeometric function of unit argument and the Landau constants

TL;DR

This paper derives convergent inverse factorial expansions for the partial sums of the Gauss hypergeometric function at unity, with results depending on the parametric excess, and connects to Landau constants.

Contribution

It introduces new convergent expansions for hypergeometric sums that depend on the parametric excess's position in the complex plane.

Findings

Expansions are valid for all n ≥ 1.

Leading behavior matches previous asymptotic results.

Special case relates to Landau constants.

Abstract

We obtain convergent inverse factorial expansions for the sum $S_n(a,b;c)$ of the first $n$ terms of the Gauss hypergeometric function of unit argument valid for $n\geq 1$. The form of these expansions depends on the location of the parametric excess $s:=c-a-b$ in the complex $s$-plane. The leading behaviour as $n\rightarrow\infty$ agrees with previous results in the literature. The case $a=b=1/2$, $c=1$ corresponds to the Landau contants.