Transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions
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- by Alfredo Deaño and Javier Segura PDF
- Math. Comp. 76 (2007), 879-901 Request permission
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.References
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Additional Information
- Alfredo Deaño
- Affiliation: Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911-Leganés (Madrid), Spain
- Email: adeano@math.uc3m.es
- Javier Segura
- Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005-Santander, Spain
- MR Author ID: 627158
- Email: javier.segura@unican.es
- Received by editor(s): February 1, 2006
- Received by editor(s) in revised form: March 24, 2006
- Published electronically: January 10, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 76 (2007), 879-901
- MSC (2000): Primary 33C05, 33C15, 39A11, 40A15, 65D20
- DOI: https://doi.org/10.1090/S0025-5718-07-01934-5
- MathSciNet review: 2291841