A Note on the 2F1 Hypergeometric Function

TL;DR

This paper revisits the convergence properties of the hypergeometric function $_{2}F_{1}$, providing new proofs and results on its convergence at specific points using innovative mathematical methods.

Contribution

It introduces a novel approach to analyze the convergence of $_{2}F_{1}$, especially at the endpoint $x=-1$, expanding understanding of its behavior for integer parameters.

Findings

Reproved convergence of $_{2}F_{1}$ for |x|<1

Established new convergence results at x=-1 for integer alpha

Developed a new theoretical framework for analyzing hypergeometric series

Abstract

The special case of the hypergeometric function $_{2}F_{1}$ represents the binomial series $(1+x)^{\alpha}=\sum_{n=0}^{\infty}(\:\alpha n\:)x^{n}$ that always converges when $|x|<1$. Convergence of the series at the endpoints, $x=\pm 1$, depends on the values of $\alpha$ and needs to be checked in every concrete case. In this note, using new approach, we reprove the convergence of the hypergeometric series $_{2}F_{1}(\alpha,\beta;\beta;x)$ for $|x|<1$ and obtain new result on its convergence at point $x=-1$ for every integer $\alpha\neq 0$. The proof is within a new theoretical setting based on the new method for reorganizing the integers and on the regular method for summation of divergent series.