Asymptotics of a Gauss hypergeometric function with large parameters, IV: A uniform expansion

We consider the uniform asymptotic expansion for the Gauss hypergeometric function $$F(a+\epsilon\lambda,m;c+\lambda;x),\qquad \lambda\to+\infty$$ for $x<1$ and positive integer $m$ when the parameter $\epsilon>1$ and the constants $a$ and $c$ are supposed finite. When $m=1$, we employ the standard procedure of the method of steepest descents modified to deal with the situation when a saddle point is near a simple pole. It is shown that it is possible to give a closed-form expression for the coefficients in the resulting uniform expansion. The expansion when $m\geq 2$ is obtained by means of a recurrence relation. Numerical results illustrating the accuracy of the resulting expansion are given.