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Identifying minimal and dominant solutions for Kummer recursions


Authors: Alfredo Deaño, Javier Segura and Nico M. Temme
Journal: Math. Comp. 77 (2008), 2277-2293
MSC (2000): Primary 33C15, 39A11, 41A60, 65D20
DOI: https://doi.org/10.1090/S0025-5718-08-02122-4
Published electronically: May 14, 2008
MathSciNet review: 2429885
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Abstract: We identify minimal and dominant solutions of three-term recurrence relations for the confluent hypergeometric functions $ _1F_1(a+\epsilon_1 n;c+\epsilon_2 n;z)$ and $ U(a+\epsilon_1 n,c+\epsilon_2 n,z)$, where $ \epsilon_i=0,\pm 1$ (not both equal to 0). The results are obtained by applying Perron's theorem, together with uniform asymptotic estimates derived by T. M. Dunster for Whittaker functions with large parameter values. The approximations are valid for complex values of $ a$, $ c$ and $ z$, with $ \vert\arg\,z\vert<\pi$.


References [Enhancements On Off] (What's this?)

  • 1. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, U.S. Government Printing Office, Washington, 1964. MR 29:4914
  • 2. T. M. Dunster, Uniform asymptotic expansions for Whittaker's confluent hypergeometric functions, SIAM J. Math. Anal. 20 (1989), no. 3, 744-760. MR 990876 (90e:33012)
  • 3. W. Gautschi, Computational aspects of three-term recurrence relations, SIAM Review 9 (1967), no. 1, 26-82. MR 0213062 (35:3927)
  • 4. A. Gil, J. Segura, and N. M. Temme, Numerical methods for special functions, SIAM, Philadelphia, PA, 2007.
  • 5. A. Gil, J. Segura, and N. M. Temme, The ABC of hyper recursions, J. Comp. Appl. Math 190 (2006), no. 1, 270-286. MR 2209508 (2006m:33003)
  • 6. -, Numerically satisfactory solutions of hypergeometric recursions, Math. Comp 76 (2007), no. 259, 1449-1468. MR 2299782
  • 7. L. J. Slater, Confluent hypergeometric functions, Cambridge University Press, New York, 1960. MR 0107026 (21:5753)
  • 8. N. M. Temme, Special functions, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1996, An introduction to the classical functions of mathematical physics. MR 97e:33002
  • 9. N. M. Temme, Uniform asymptotic expansions of Laplace integrals, Analysis 3 (1983), 221-249. MR 756117 (85j:41059)

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Additional Information

Alfredo Deaño
Affiliation: DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, CB3 0WA, United Kingdom
Email: ad495@cam.ac.uk

Javier Segura
Affiliation: Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, 39005 Santander, Spain
Email: javier.segura@unican.es

Nico M. Temme
Affiliation: CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands
Email: nicot@cwi.nl

DOI: https://doi.org/10.1090/S0025-5718-08-02122-4
Keywords: Kummer functions, Whittaker functions, confluent hypergeometric functions, recurrence relations, difference equations, stability of recurrence relations, numerical evaluation of special functions, asymptotic analysis
Received by editor(s): August 30, 2007
Published electronically: May 14, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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