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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.