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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Vydané v:	Numerische Mathematik Ročník 116; číslo 2; s. 269 - 289
Hlavní autori:	López, José Luis, Temme, Nico M.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer-Verlag 01.08.2010 Springer
Predmet:	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
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Abstract	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.
AbstractList	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.
Author	Temme, Nico M. López, José Luis
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Keywords	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
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References	López, Temme (CR10) 2010; 363 Gil, Segura, Temme (CR6) 2007 CR4 Tricomi (CR15) 1947; 26 Maino, Menapace, Ventura (CR11) 1981; 40 CR7 CR16 Abramowitz, Stegun (CR3) 1964 CR12 López, Temme (CR9) 2004; 356 López, Temme (CR8) 2002; 109 Temme (CR14) 1996 Abad, Sesma (CR2) 1997; 78 Slater (CR13) 1960 Chiccoli, Lorenzutta, Maino (CR5) 1989; 103 Abad, Sesma (CR1) 1999; 101 J.L. López (303_CR8) 2002; 109 303_CR9 A. Gil (303_CR6) 2007 J. Abad (303_CR1) 1999; 101 J. Abad (303_CR2) 1997; 78 L.J. Slater (303_CR13) 1960 303_CR12 J.L. López (303_CR10) 2010; 363 303_CR16 M. Abramowitz (303_CR3) 1964 C. Chiccoli (303_CR5) 1989; 103 N.M. Temme (303_CR14) 1996 303_CR4 303_CR7 G. Maino (303_CR11) 1981; 40 F. Tricomi (303_CR15) 1947; 26
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SubjectTerms	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
Title	Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions
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