On the stability of recurrence relations for hypergeometric functions

Abstract

We consider the three term recurrence relations y_n+1 + a_n y_n + b_n y_n-1 = 0 satisfied simultaneously by confluent hypergeometric functions M(a+kn; c+mn; x) and U(a+kn; c+mn; x) (up to normalizations not depending on x). The parameters a, c, x are fixed and k,m = 0,±1. The existence of minimal solutions when n -> ∞ is a crucial piece of information when we intend to use a recurrence relation for computation. However, in some cases the behavior of the solutions for moderate values of n can be opposite to the asymptotic behaviour. We provide numerical examples of this phenomenon, already noted by W. Gautschi in the case (k,m) = (1,1), both for the recurrence relations and for the associated continued fractions.