×

Asymptotics of some generalized Mathieu series. (English) Zbl 1455.33015

Let \(\mu\ge0\) and \(r>0\), and fix two series \(\mathbf{a}=(a_n)_{n\ge0}\), \(\mathbf{b}=(b_n)_{n\ge0}\). The generalized Mathieu series is defined as \[S_{\mathrm{a}, \mathrm{b}, \mu}(r):=\sum_{n=0}^{\infty} \frac{a_{n}}{\left(b_{n}+r^{2}\right)^{\mu+1}}.\] Moreover, for \(\alpha, \beta, r>0, \mu \geq 0,\) with \(\alpha-\beta(\mu+1)<-1\) and \(\gamma, \delta \in \mathbb{R}\) the series \[ S_{\alpha, \beta, \gamma, \delta, \mu}(r):=\sum_{n=2}^{\infty} \frac{n^{\alpha}(\log n)^{\gamma}}{\left(n^{\beta}(\log n)^{\delta}+r^{2}\right)^{\mu+1}} \] is defined. Another special case of the generalized Mathieu series is \[S_{\alpha, \beta, \mu}^{!}(r):=\sum_{n=0}^{\infty} \frac{(n !)^{\alpha}}{\left((n !)^{\beta}+r^{2}\right)^{\mu+1}},\] for appropriate parameter choice.
The authors then study the asymptotic behavior of these series. Among others, it turns out that \[S_{\alpha, \beta, \gamma, \delta, \mu}(r) \sim C_{\alpha, \beta, \gamma, \delta, \mu} r^{2(\alpha+1) / \beta-2(\mu+1)}(\log r)^{-\delta(\alpha+1) / \beta+\gamma}, \quad r \uparrow \infty,\] again, for appropriate parameter choice. Similar asymptotics hold for \(S_{\alpha, \beta, \mu}^{!}(r)\), too. Here \(C_{\alpha, \beta, \gamma, \delta, \mu}\) is an expression involving the Gamma function.

MSC:

33E20 Other functions defined by series and integrals
11M41 Other Dirichlet series and zeta functions
44A15 Special integral transforms (Legendre, Hilbert, etc.)

Software:

DLMF
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bingham, N. H., Goldie, C. M., and Teugels, J. L., Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. https://doi.org/10.1017/CBO9780511721434 · Zbl 0617.26001
[2] Borwein, J. M. and Corless, R. M., Gamma and factorial in the Monthly, Amer. Math. Monthly 125 (2018), no. 5, 400-424. https://doi.org/10.1080/00029890.2018.1420983 · Zbl 1392.33001
[3] Copson, E. T., An introduction to the theory of functions of a complex variable, Clarendon Press, Oxford, 1935. · Zbl 0012.16902
[4] Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., and Knuth, D. E., On the Lambert \(W\) function, Adv. Comput. Math. 5 (1996), no. 4, 329-359. https://doi.org/10.1007/BF02124750 · Zbl 0863.65008
[5] Costin, O. and Huang, M., Behavior of lacunary series at the natural boundary, Adv. Math. 222 (2009), no. 4, 1370-1404. https://doi.org/10.1016/j.aim.2009.06.011 · Zbl 1182.30005
[6] Drmota, M. and Soria, M., Marking in combinatorial constructions: generating functions and limiting distributions, Theoret. Comput. Sci. 144 (1995), no. 1-2, 67-99. https://doi.org/10.1016/0304-3975(94)00294-S · Zbl 0874.68143
[7] Elbert, Á., Asymptotic expansion and continued fraction for Mathieu’s series, Period. Math. Hungar. 13 (1982), no. 1, 1-8. https://doi.org/10.1007/BF01848090 · Zbl 0504.41025
[8] Flajolet, P., Singularity analysis and asymptotics of Bernoulli sums, Theoret. Comput. Sci. 215 (1999), no. 1-2, 371-381. https://doi.org/10.1016/S0304-3975(98)00220-5 · Zbl 0913.68098
[9] Flajolet, P., Fusy, E., Gourdon, X., Panario, D., and Pouyanne, N., A hybrid of Darboux’s method and singularity analysis in combinatorial asymptotics, Electron. J. Combin. 13 (2006), no. 1, res. paper 103, 35 pp. · Zbl 1111.05006
[10] Flajolet, P., Gourdon, X., and Dumas, P., Mellin transforms and asymptotics: harmonic sums, Theoret. Comput. Sci. 144 (1995), no. 1-2, 3-58. https://doi.org/10.1016/0304-3975(95)00002-E · Zbl 0869.68057
[11] Flajolet, P. and Odlyzko, A., Singularity analysis of generating functions, SIAM J. Discrete Math. 3 (1990), no. 2, 216-240. https://doi.org/10.1137/0403019 · Zbl 0712.05004
[12] Flajolet, P. and Sedgewick, R., Analytic combinatorics, Cambridge University Press, Cambridge, 2009. https://doi.org/10.1017/CBO9780511801655 · Zbl 1165.05001
[13] Ford, W. B., Studies on divergent series and summability & The asymptotic developments of functions defined by Maclaurin series, Chelsea Publishing Co., New York, 1960.
[14] Friz, P. and Gerhold, S., Extrapolation analytics for Dupire’s local volatility, in “Large deviations and asymptotic methods in finance”, Springer Proc. Math. Stat., vol. 110, Springer, Cham, 2015, pp. 273-286. https://doi.org/10.1007/978-3-319-11605-1_10 · Zbl 1418.91511
[15] Gerhold, S. and Tomovski, Ž., Asymptotic expansion of Mathieu power series and trigonometric Mathieu series, J. Math. Anal. Appl. 479 (2019), no. 2, 1882-1892. https://doi.org/10.1016/j.jmaa.2019.07.029 · Zbl 1450.30006
[16] Grabner, P. J. and Thuswaldner, J. M., Analytic continuation of a class of Dirichlet series, Abh. Math. Sem. Univ. Hamburg 66 (1996), 281-287. https://doi.org/10.1007/BF02940810 · Zbl 0879.11047
[17] Hardy, G. H. and Riesz, M., The general theory of Dirichlet’s series, Cambridge Tracts in Mathematics and Mathematical Physics, no. 18, Cambridge University Press, Cambridge, 1915. · JFM 45.0387.03
[18] Mehrez, K. and Tomovski, Ž., On a new \((p,q)\)-Mathieu-type power series and its applications, Appl. Anal. Discrete Math. 13 (2019), no. 1, 309-324. https://doi.org/10.2298/AADM190427005M
[19] NIST digital library of mathematical functions, http://dlmf.nist.gov/, release 1.0.27 of 2020-06-15, F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds.
[20] Olver, F. W. J., Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press, New York, London, 1974. · Zbl 0303.41035
[21] Olver, F. W. J., Lozier, D. W., Boisvert, R. F., and Clark, C. W. (eds.), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. · Zbl 1198.00002
[22] Paris, R. B., The discrete analogue of Laplace’s method, Comput. Math. Appl. 61 (2011), no. 10, 3024-3034. https://doi.org/10.1016/j.camwa.2011.03.092 · Zbl 1222.41043
[23] Paris, R. B., The asymptotic expansion of a generalised Mathieu series, Appl. Math. Sci. (Ruse) 7 (2013), no. 125-128, 6209-6216. https://doi.org/10.12988/ams.2013.39492
[24] Paris, R. B., Asymptotic expoansions of Mathieu-Bessel series. I, eprint arXiv:1907.01812 [math.CA], \nooptsort 2019a2019.
[25] Paris, R. B., Asymptotic expansion of Mathieu-Bessel series. II, eprint arXiv:1909.09805 [math.CA], \nooptsort 2019b2019.
[26] Pogány, T. K., Srivastava, H. M., and Tomovski, v., Some families of Mathieu \(\mathbf a\)-series and alternating Mathieu \(\mathbf a\)-series, Appl. Math. Comput. 173 (2006), no. 1, 69-108. https://doi.org/10.1016/j.amc.2005.02.044 · Zbl 1097.33016
[27] Srivastava, H. M., Sums of certain series of the Riemann zeta function, J. Math. Anal. Appl. 134 (1988), no. 1, 129-140. https://doi.org/10.1016/0022-247X(88)90013-3 · Zbl 0632.10040
[28] Srivastava, H. M., Mehrez, K., and Tomovski, Ž., New inequalities for some generalized Mathieu type series and the Riemann zeta function, J. Math. Inequal. 12 (2018), no. 1, 163-174. https://doi.org/10.7153/jmi-2018-12-13 · Zbl 1391.33007
[29] Srivastava, H. M. and Tomovski, Ž., Some problems and solutions involving Mathieu’s series and its generalizations, JIPAM. J. Inequal. Pure Appl. Math. 5 (2004), no. 2, article 45, 13 pp. · Zbl 1068.33032
[30] Tomovski, Ž., Some new integral representations of generalized Mathieu series and alternating Mathieu series, Tamkang J. Math. 41 (2010), no. 4, 303-312. · Zbl 1220.33021
[31] Tomovski, Ž. and Pogány, T. K., Integral expressions for Mathieu-type power series and for the Butzer-Flocke-Hauss Ω-function, Fract. Calc. Appl. Anal. 14 (2011), no. 4, 623-634. https://doi.org/10.2478/s13540-011-0036-2 · Zbl 1273.33016
[32] Whittaker, E. T. and Watson, G. N., A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996, reprint of fourth edition, 1927. https://doi.org/10.1017/CBO9780511608759 · JFM 45.0433.02
[33] Zastavny\u ı, V. P., Asymptotic expansion of some series and their application, Ukr. Mat. Visn. 6 (2009), no. 4, 553-573.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.