Identifying minimal and dominant solutions for Kummer recursions

Abstract:

We identify minimal and dominant solutions of three-term recurrence relations for the confluent hypergeometric functions $_1F_1(a+\epsilon _1 n;c+\epsilon _2 n;z)$ and $U(a+\epsilon _1 n,c+\epsilon _2 n,z)$, where $\epsilon _i=0,\pm 1$ (not both equal to 0). The results are obtained by applying Perron’s theorem, together with uniform asymptotic estimates derived by T. M. Dunster for Whittaker functions with large parameter values. The approximations are valid for complex values of $a$, $c$ and $z$, with $|\arg z|<\pi$.