Asymptotics of some generalized Mathieu series
DOI:
https://doi.org/10.7146/math.scand.a-121106Abstract
We establish asymptotic estimates of Mathieu-type series defined by sequences with power-logarithmic or factorial behavior. By taking the Mellin transform, the problem is mapped to the singular behavior of certain Dirichlet series, which is then translated into asymptotics for the original series. In the case of power-logarithmic sequences, we obtain precise first order asymptotics. For factorial sequences, a natural boundary of the Mellin transform makes the problem more challenging, but a direct elementary estimate gives reasonably precise asymptotics. As a byproduct, we prove an expansion of the functional inverse of the gamma function at infinity.
References
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
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
Copson, E. T., An introduction to the theory of functions of a complex variable, Clarendon Press, Oxford, 1935.
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
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
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
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
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
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.
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
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
Flajolet, P. and Sedgewick, R., Analytic combinatorics, Cambridge University Press, Cambridge, 2009. https://doi.org/10.1017/CBO9780511801655
Ford, W. B., Studies on divergent series and summability & The asymptotic developments of functions defined by Maclaurin series, Chelsea Publishing Co., New York, 1960.
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
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
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
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.
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
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.
Olver, F. W. J., Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press, New York, London, 1974.
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.
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
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
Paris, R. B., Asymptotic expoansions of Mathieu-Bessel series. I, eprint arXiv:1907.01812 [math.CA], \nooptsort 2019a2019.
Paris, R. B., Asymptotic expansion of Mathieu-Bessel series. II, eprint arXiv:1909.09805 [math.CA], \nooptsort 2019b2019.
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
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
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
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.
Tomovski, Ž., Some new integral representations of generalized Mathieu series and alternating Mathieu series, Tamkang J. Math. 41 (2010), no. 4, 303–312.
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
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
Zastavny\u ı, V. P., Asymptotic expansion of some series and their application, Ukr. Mat. Visn. 6 (2009), no. 4, 553–573.
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
Copson, E. T., An introduction to the theory of functions of a complex variable, Clarendon Press, Oxford, 1935.
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
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
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
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
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
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.
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
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
Flajolet, P. and Sedgewick, R., Analytic combinatorics, Cambridge University Press, Cambridge, 2009. https://doi.org/10.1017/CBO9780511801655
Ford, W. B., Studies on divergent series and summability & The asymptotic developments of functions defined by Maclaurin series, Chelsea Publishing Co., New York, 1960.
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
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
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
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.
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
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.
Olver, F. W. J., Asymptotics and special functions, Computer Science and Applied Mathematics, Academic Press, New York, London, 1974.
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.
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
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
Paris, R. B., Asymptotic expoansions of Mathieu-Bessel series. I, eprint arXiv:1907.01812 [math.CA], \nooptsort 2019a2019.
Paris, R. B., Asymptotic expansion of Mathieu-Bessel series. II, eprint arXiv:1909.09805 [math.CA], \nooptsort 2019b2019.
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
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
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
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.
Tomovski, Ž., Some new integral representations of generalized Mathieu series and alternating Mathieu series, Tamkang J. Math. 41 (2010), no. 4, 303–312.
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
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
Zastavny\u ı, V. P., Asymptotic expansion of some series and their application, Ukr. Mat. Visn. 6 (2009), no. 4, 553–573.
Downloads
Published
2020-09-03
How to Cite
Gerhold, S., Hubalek, F., & Tomovski, Živorad. (2020). Asymptotics of some generalized Mathieu series. MATHEMATICA SCANDINAVICA, 126(3), 424–450. https://doi.org/10.7146/math.scand.a-121106
Issue
Section
Articles
