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On the stability of recurrence relations for hypergeometric functions

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URI: https://hdl.handle.net/10016/6657
ISBN: 3-527-40652-2
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stability_deano_pre_2005_ps.pdf (19.41 KB)
recurrences_deano_icnaam_2005_ps.pdf (30.71 KB)
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2005
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Wiley VCH
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Abstract
We consider the three term recurrence relations y_n+1 + a_n y_n + b_n y_n-1 = 0 satisfied simultaneously by confluent hypergeometric functions M(a+kn; c+mn; x) and U(a+kn; c+mn; x) (up to normalizations not depending on x). The parameters a, c, x are fixed and k,m = 0,±1. The existence of minimal solutions when n -> ∞ is a crucial piece of information when we intend to use a recurrence relation for computation. However, in some cases the behavior of the solutions for moderate values of n can be opposite to the asymptotic behaviour. We provide numerical examples of this phenomenon, already noted by W. Gautschi in the case (k,m) = (1,1), both for the recurrence relations and for the associated continued fractions.
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4 pages, no figures.-- MSC2000 codes: 33C15, 33F05, 40A15.-- Running title: "Recurrences and continued fractions for Kummer functions".
Contributed to: ICNAAM 2005: Official conference of the European Society of Computational Methods in Sciences and Engineering (Rhodes, Greece, Sep 16-20, 2005).
Zbl#: Zbl 1086.33007
Keywords
Confluent hypergeometric functions, Continued fractions, Three-term recurrence relations
Bibliographic citation
Simos, Theodore S. (ed.) et al., ICNAAM 2005: International conference on numerical analysis and applied mathematics 2005, p. 672-675
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