The Gauss hypergeometric function F(a,b;c;z)F(a,b;c;z) for large c

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