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Mathematics of Computation

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

Authors: Alfredo Deaño and Javier Segura
Journal: Math. Comp. 76 (2007), 879-901
MSC (2000): Primary 33C05, 33C15, 39A11, 40A15, 65D20
Published electronically: January 10, 2007
MathSciNet review: 2291841
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Abstract: Three term recurrence relations $ y_{n+1}+b_n y_n+a_n y_{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 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 $ 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 implication of such transitory behaviour on the numerical stability of recursion.

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Additional Information

Alfredo Deaño
Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911-Leganés (Madrid), Spain

Javier Segura
Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005-Santander, Spain

Keywords: Hypergeometric functions, recurrence relations, condition and stability, continued fractions, numerical evaluation of special functions
Received by editor(s): February 1, 2006
Received by editor(s) in revised form: March 24, 2006
Published electronically: January 10, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.