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

Three term recurrence relations yn+1 +bnyn +anyn?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 role 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 an negative and bn 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 implication of such

transitory behaviour on the numerical stability of recursion.