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

We obtain convergent inverse factorial expansions for the sum Sn(a, b; c) of the first n ≥ 1 terms of the Gauss hypergeometric function ₂F₁(a, b; c; 1) of unit argument. 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 → ∞ agrees with previous results in the literature. The case a = b = 1/2 , c = 1 corresponds to the Landau contants for which an expansion is obtained.