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

Deaño, Alfredo, Segura, Javier (2007) Transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions. Mathematics of Computation, 76 (258). pp. 879-901. ISSN 0025-5718. (doi:10.1090/S0025-5718-07-01934-5) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

Abstract

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.

Item Type:	Article
DOI/Identification number:	10.1090/S0025-5718-07-01934-5
Uncontrolled keywords:	Hypergeometric functions, recurrence relations, condition and stability, continued fractions, numerical evaluation of special functions.
Subjects:	Q Science > QA Mathematics (inc Computing science) Q Science > QA Mathematics (inc Computing science) > QA297 Numerical analysis Q Science > QA Mathematics (inc Computing science) > QA351 Special functions
Divisions:	Faculties > Sciences > School of Mathematics Statistics and Actuarial Science > Applied Mathematics
Depositing User:	Alfredo Deano-Cabrera
Date Deposited:	21 Nov 2018 10:46 UTC
Last Modified:	30 May 2019 08:21 UTC
Resource URI:	https://kar.kent.ac.uk/id/eprint/70230 (The current URI for this page, for reference purposes)
Deaño, Alfredo:	https://orcid.org/0000-0003-1704-247X