Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions

Expansions in terms of Bessel functions are considered of the Kummer function 1 F 1 ( a ; c , z ) (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials in the parameters of the Kummer function and the asymptotic behavior of thes...

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Bibliographic Details
Published in:Numerische Mathematik Vol. 116; no. 2; pp. 269 - 289
Main Authors: López, José Luis, Temme, Nico M.
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.08.2010
Springer
Subjects:
Difference and functional equations, recurrence relations
Exact sciences and technology
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical analysis. Scientific computation
Numerical and Computational Physics
Sciences and techniques of general use
Simulation
Special functions
Theoretical
Asymptotic expansions
33C15
65D20
33C10
Kummer functions
41A30
Confluent hypergeometric functions
41A60
Computation of special functions
Bessel functions
Bessel function
Polynomial
Transcendental equation
Numerical analysis
Asymptotic expansion
Recurrence relation
Asymptotic behavior
Hypergeometric function
Asymptotic approximation
Difference equation
Numerical stability
ISSN:0029-599X, 0945-3245
Online Access:Get full text
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Summary:Expansions in terms of Bessel functions are considered of the Kummer function 1 F 1 ( a ; c , z ) (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials in the parameters of the Kummer function and the asymptotic behavior of these polynomials for large degree is given. Tables are given to show the rate of approximation of the asymptotic estimates. The numerical performance of the expansions is discussed together with the numerical stability of recurrence relations to compute the polynomials. The asymptotic character of the expansions is explained for large values of the parameter a of the Kummer function.
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-010-0303-x