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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$.


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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.