zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions. (English) Zbl 1195.65025
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 F. Tricomi [Ann. Mat. Pura Appl., IV. Ser. 26, 141–175 (1947; Zbl 0034.33704)] and 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:
65D20Computation of special functions, construction of tables
References:
[1]Abad J., Sesma J.: Buchholz polynomials: a family of polynomials relating solutions of confluent hypergeometric and Bessel equations. J. Comput. Appl. Math. 101(1–2), 237–241 (1999) · Zbl 0940.33001 · doi:10.1016/S0377-0427(99)00226-5
[2]Abad J., Sesma J.: A new expansion of the confluent hypergeometric function in terms of modified Bessel functions. J. Comput. Appl. Math. 78(1), 97–101 (1997) · Zbl 0931.33002 · doi:10.1016/S0377-0427(96)00133-1
[3]Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol 55 of National Bureau of Standards Applied Mathematics Series. U.S. Government Printing Office, Washington (1964)
[4]Buchholz, H.: The Confluent Hypergeometric Function with Special Emphasis on Its Applications. Translated from the German by Lichtblau, H., Wetzel, K. Springer Tracts in Natural Philosophy, vol. 15. Springer, New York (1969)
[5]Chiccoli C., Lorenzutta S., Maino G.: A note on a Tricomi expansion for the generalized exponential integral and related functions. Nuovo Cimento B (11) 103(5), 563–568 (1989) · doi:10.1007/BF02753139
[6]Gil A., Segura J., Temme N.M.: Numerical Methods for Special Functions. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2007)
[7]Kreuser, P.: Über das Verhalten der Integrale homogener linearer Differenzengleichungen im Unendlichen. PhD thesis, Diss. Tübingen, 48S (1914)
[8]López J.L., Temme N.M.: Two-point Taylor expansions of analytic functions. Stud. Appl. Math. 109(4), 297–311 (2002) · Zbl 1141.30300 · doi:10.1111/1467-9590.00225
[9]López, J.L., Temme, N.M.: Multi-point Taylor expansions of analytic functions. Trans. Amer. Math. Soc. 356(11), 4323–4342 (electronic) (2004)
[10]López J.L., Temme N.M.: Large degree asymptotics of generalized Bernoulli and Euler polynomials. J. Math. Anal. Appl. 363(1), 197–208 (2010) · Zbl 1213.11051 · doi:10.1016/j.jmaa.2009.08.034
[11]Maino G., Menapace E., Ventura A.: Computation of parabolic cylinder functions by means of a Tricomi expansion. J. Comput. Phys. 40(2), 294–304 (1981) · Zbl 0466.65016 · doi:10.1016/0021-9991(81)90211-4
[12]Olver, F.W.J.: Asymptotics and Special Functions. AKP Classics. A K Peters Ltd., Wellesley (1997). Reprint, with corrections, of original Academic Press edition (1974)
[13]Slater L.J.: Confluent Hypergeometric Functions. Cambridge University Press, New York (1960)
[14]Temme N.M.: Special Functions. An Introduction to the Classical Functions of Mathematical Physics. A Wiley-Interscience Publication. Wiley, New York (1996)
[15]Tricomi F.: Sulle funzioni ipergeometriche confluenti. Ann. Mat. Pura Appl. (4) 26, 141–175 (1947) · Zbl 0034.33704 · doi:10.1007/BF02415375
[16]Wong, R.: Asymptotic Approximations of Integrals, vol. 34 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Corrected reprint of the 1989 original (2001)