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Zbl 1195.65025
López, José Luis; Temme, Nico M.
Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions. (English)
[J] Numer. Math. 116, No. 2, 269-289 (2010). ISSN 0029-599X; ISSN 0945-3245/e

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 {\it F. Tricomi} [Ann. Mat. Pura Appl., IV. Ser. 26, 141--175 (1947; Zbl 0034.33704)] and {\it H. Buchholz} [The confluent hypergeometric function with special emphasis on its applications. Berlin-Heidelberg-New York: Springer-Verlag (1969; Zbl 0169.08501)]. 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.

MSC 2000:: *65D20 Computation of special functions

Keywords: Kummer functions; confluent hypergeometric functions; Bessel functions; asymptotic expansions; computation of special functions

Citations: Zbl 0034.33704; Zbl 0169.08501

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