A Note on the 2F1 Hypergeometric Function

The special case of the hypergeometric function 2F1 represents the binomial series (1 + x) = ∑∞ n=0 ( α n ) xn that always converges when |x| < 1. Convergence of the series at the endpoints, x = ±1, depends on the values of α and needs to be checked in every concrete case. In this note, using new approach, we reprove the convergence of the hypergeometric series for |x| < 1 and obtain new result on its convergence at point x = −1 for every integer α 0, that is we prove it for the function 2F1(α, β; β; x). The proof is within a new theoretical setting based on a new method for reorganizing the integers and on the original regular method for summation of divergent series.