Asymptotics of the Gauss hypergeometric function with large parameters, I

We obtain asymptotic expansions for the Gauss hypergeometric function F(a+ε1λ,b+ε2λ;c+ε3λ;z) as |λ| →∞ when the εj are finite by an application of the method of steepest descents, thereby extending previous results corresponding to εj = 0, ±1. By means of connection formulas satisfied by F it is possible to arrange the above hypergeometric function into three basic groups. In Part I we consider the cases (i) ε1 > 0, ε2 = 0, ε3 = 1 and (ii) ε1 > 0, ε2 = −1, ε3 = 0; the third case ε1, ε2 >0, ε3 =1 is deferred to Part II. The resulting expansions are of Poincar´e type and hold in restricted domains of the complex z-plane. Numerical results illustrating the accuracy of the different expansions are given.