Transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions
Alfredo Deaño and Javier Segura
Math. Comp. 76 (2007), 879-901
Primary 33C05, 33C15, 39A11, 40A15, 65D20
January 10, 2007
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Abstract: Three term recurrence relations 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 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 negative and 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.
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Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911-Leganés (Madrid), Spain
Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005-Santander, Spain
condition and stability,
numerical evaluation of special functions
Received by editor(s):
February 1, 2006
Received by editor(s) in revised form:
March 24, 2006
January 10, 2007
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American Mathematical Society
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