Uniform asymptotics of a Gauss hypergeometric function with two large parameters, V (2004.01945v2)

Abstract: We consider the uniform asymptotic expansion for the Gauss hypergeometric function [{}_2F_1(a+\epsilon\lambda,b;c+\lambda;x),\qquad 0<x\<1\] as $\lambda\to+\infty$ in the neigbourhood of $\epsilon x=1$ when the parameter $\epsilon\>1$ and the constants $a$, $b$ and $c$ are supposed finite. Use of a standard integral representation shows that the problem reduces to consideration of a simple saddle point near an endpoint of the integration path. A uniform asymptotic expansion is first obtained by employing Bleistein's method. An alternative form of uniform expansion is derived following the approach described in Olver's book [{\it Asymptotics and Special Functions}, p.~346]. This second form has several advantages over the Bleistein form.