The Gauss hypergeometric function for large c

We consider the Gauss hypergeometric function F ( a , b + 1 ; c + 2 ; z ) for a , b , c ∈ C , c ≠ - 2 ,- 3 - 4 , … and | arg ( 1 - z ) | < π . We derive a convergent expansion of F ( a , b + 1 ; c + 2 ; z ) in terms of rational functions of a, b, c and z valid for | b | | z | < | c - bz | and | c - b | | z | < | c - bz | . This expansion has the additional property of being asymptotic for large c with fixed a uniformly in b and z (with bounded b / c ). Moreover, the asymptotic character of the expansion holds for a larger set of b, c and z specified below.