Transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions

Abstract

Three-term recurrence relations $y_{n+1}+b_ny_n+a_ny_{n-1}=0$ can be used for computing, recursively, a great number of special functions. Depending on the asymptotic nature of the function to be computed, different recursion directions need to be considered: backward for minimal solutions and forward for dominant solutions. However, some solutions interchange their roles for finite values of $n$ with respect to their asymptotic behaviour and certain dominant solutions may transitorily behave as minimal. This phenomenon, related to Gautschi's anomalous convergence of the continued fraction for ratios of confluent hypergeometric functions, is shown to be a general situation which takes place for recurrences with $a_n$ negative and $b_n$ changing sign once. We analyze the anomalous convergence of the associated continued fractions for a number of different recurrence relations (modified Bessel functions, confluent and Gauss hypergeometric functions) and discuss the implications of such transitory behaviour for the numerical stability of recursion.