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Asymptotic and numerical aspects of the generalized Marcum function of the second kind. (English) Zbl 1499.41095

Summary: The asymptotic and numerical behavior of the so-called generalized Marcum function of the second kind is considered. By using some known expansions on the modified Bessel function of the second kind, we deduce the asymptotic expansions for the generalized Marcum function of the second kind for large parameters, and all the expansions are obtained when exactly one parameter is large and others are fixed. We also show numerically that the asymptotic formulas obtained in this paper are good approximations.

MSC:

41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
30E05 Moment problems and interpolation problems in the complex plane
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)

Software:

DLMF; Algorithm 939
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Full Text: DOI

References:

[1] Á. Baricz, N. Bisht, S. Singh, V.A. Vijesh: The generalized Marcum function of the second kind: monotonicity patterns and tight bounds., J. Comput. Appl. Math, 382 (2021), Art. 113093. · Zbl 1472.33015
[2] Á. Baricz, N. Bisht, S. Singh, V.A. Vijesh: Bounds for the generalized Marcum function of the second kind., Ramanujan J., 58 (2022), 1-21. doi: 10.1007/s11139-021-00440-9. · Zbl 1509.33023 · doi:10.1007/s11139-021-00440-9
[3] A. Gil, J. Segura, N.M. Temme: Algorithm 939: Computation of the Marcum Q-function. ACM Trans. Math. Software, 40 (3) (2014), Art. 20. · Zbl 1322.65046
[4] S. Nadarajah: A modified Bessel distribution of the second kind., Statistica, 47 (4) (2007), 405-413. · Zbl 1189.62014
[5] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (Eds.): NIST Handbook of Mathematical Functions. Cambridge Univ. Press, Cambridge, 2010. · Zbl 1198.00002
[6] R. Wong: Asymptotic approximations of integrals., Academic Press, 1989. · Zbl 0679.41001
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