Schlömilch Series

  • Árpád Baricz
  • Dragana Jankov Maširević
  • Tibor K. Pogány
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2207)

Abstract

This chapter is devoted to the study of integral representations of Schlömilch series built by Bessel functions of the first kind and modified Bessel functions of the second kind. Closed expressions for some special Schlömilch series together with their connection to Mathieu series are also investigated. The chapter ends with an integral representation formula for number theoretical summation by Popov, which also covers the theta-transform identity coming from functional equation for the Epstein Zeta function.

References

  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1965)Google Scholar
  2. 4.
    Al-Jarrah, A., Dempsey, K.M., Glasser, M.L.: Generalized series of Bessel functions. J. Comput. Appl. Math. 143, 1–8 (2002)Google Scholar
  3. 28.
    Berndt, B.C., Kim, S.: Identities for logarithmic means: a survey. In: Alaca, A., Alaca, Ş., Williams, K.S. (eds.) Advances in the Theory of Numbers. Proceedings of the Thirteenth Conference of the Canadian Number Theory Association, Carleton University, Ottawa, ON (June 16–20, 2014). Fields Institute Communications, vol. 77. Fields Institute for Research in Mathematical Sciences, Toronto, ON (2015)Google Scholar
  4. 30.
    Berndt, B.C., Zaharescu, A.: Weighted divisor sums and Bessel function series. Math. Ann. 335(2), 249–283 (2006)Google Scholar
  5. 31.
    Berndt, B.C., Kim, S., Zaharescu, A.: Weighted divisor sums and Bessel function series II. Adv. Math. 229(3), 2055–2097 (2012)Google Scholar
  6. 32.
    Berndt, B.C., Kim, S., Zaharescu, A.: Weighted divisor sums and Bessel function series IV. Ramanujan J. 29(1–3), 79–102 (2012)Google Scholar
  7. 33.
    Berndt, B.C., Kim, S., Zaharescu, A.: Weighted divisor sums and Bessel function series III. J. Reine Angew. Math. 683, 67–96 (2013)Google Scholar
  8. 34.
    Berndt, B.C., Kim, S., Zaharescu, A.: Weighted divisor sums and Bessel function series V. J. Approx. Theory 197, 101–114 (2015)Google Scholar
  9. 35.
    Berndt, B.C., Dixit, A., Kim, S., Zaharescu, A.: On a theorem of A. I. Popov on sum of squares. Proc. Amer. Math. Soc. 145(9), 3795–3808 (2017). https://doi.org/10.1090/proc/13547Google Scholar
  10. 38.
    Bondarenko, V.F.: Efficient summation of Schlömilch series of cylindrical functions. USSR Comput. Math. Math. Phys. 31(7), 101–104 (1991)Google Scholar
  11. 48.
    Chandrasekharan, K., Narasimhan, R.: Hecke’s functional equation and arithmetical identities. Ann. Math. 74, 1–23 (1961)Google Scholar
  12. 50.
    Chaudhry, M.A., Qadir, A., Boudjelkha, M.T., Rafique, M., Zubair, S.M.: Extended Riemann zeta functions. Rocky Mt. J. Math. 31(4), 1237–1263 (2001)Google Scholar
  13. 53.
    Cerone, P., Lenard, C.T.: On integral forms of generalized Mathieu series. J. Inequal. Pure Appl. Math. 4(5), Art. 100, 1–11 (2003)Google Scholar
  14. 55.
    Coates, C.V.: Bessel’s functions of the second order. Q. J. XXI, 183–192 (1886)Google Scholar
  15. 63.
    Diethelm, K., Ford, N.J., Freed, A.D., Luchko, Yu.: Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194, 743–773 (2005)Google Scholar
  16. 76.
    Epstein, P.: Zur Theorie allgemeiner Zetafunktionen I. Math. Ann. 56, 614–644 (1903)Google Scholar
  17. 77.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York, Toronto, London (1953)Google Scholar
  18. 83.
    Filon, L.N.G.: On the expansion of polynomials in series of functions. Proc. Lond. Math. Soc. (Ser. 2). IV, 396–430 (1907)Google Scholar
  19. 93.
    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 6th edn. Academic, San Diego, CA (2000)Google Scholar
  20. 105.
    Hardy, G.H., Wright, E.M.: In: Heath-Brown, D.R., Silverman, J.H. (eds.) An Introduction to the Theory of Numbers, 6th edn. Oxford Science Publications/Clarendon Press, Oxford/London (2008). “The Function r(n),” “Proof of the Formula for r(n)”, “The Generating Function of r(n)”, “The Order of r(n)”, and “Representations by a Larger Number of Squares.” §16.9, 16.10, 17.9, 18.7, and 20.13, pp. 241–243, 256–258, 270–271, 314–315Google Scholar
  21. 108.
    Hilbert, D., Cohn-Vossen, S.: Geometry and the Imagination. Chelsea Publishing Company, New York (1952)Google Scholar
  22. 117.
  23. 119.
  24. 122.
  25. 131.
    Jankov, D., Pogány, T.K.: Integral representation of Schlömilch series. J. Class. Anal. 1(1), 75–84 (2012)Google Scholar
  26. 135.
    Jankov Maširević, D.: Summations of Schlömilch series containing modified Bessel function of the second kind terms. Integral Transforms Spec. Funct. 26(4), 273–281 (2015)Google Scholar
  27. 140.
    Jankov Maširević, D., Pogány, T.K.: p-extended Mathieu series from the Schlömilch series point of view. Vietnam J. Math. 45(4), 713–719 (2017)Google Scholar
  28. 157.
    Koshliakov, N.S.: Sum-formulae containing numerical functions. J. Soc. Phys. Math. Leningr. 2, 53–76 (1928)Google Scholar
  29. 175.
    Lorch, L., Szego, P.: Closed expressions for some infinite series of Bessel and Struve functions. J. Math. Anal. Appl. 122, 47–57 (1987)Google Scholar
  30. 197.
    Miller, A.R.: m-dimensional Schlömilch series. Can. Math. Bull. 38(3), 347–351 (1995)Google Scholar
  31. 198.
    Miller, A.R.: On certain Schlömilch-type series. J. Comput. Appl. Math. 80, 83–95 (1997)Google Scholar
  32. 201.
    Milovanović, G.V.; Pogány, T.K.: New integral forms of generalized Mathieu series and related applications. Appl. Anal. Discrete Math. 7(1), 180–192 (2013)Google Scholar
  33. 206.
    Murio, D.A.: Stable numerical evaluation of Grünwald-Letnikov fractional derivatives applied to a fractional IHCP. Inverse Probl. Sci. Eng. 17, 229–243 (2009)Google Scholar
  34. 212.
    Nielsen, N.: Flertydige Udviklinger efter Cylinderfunktioner. Nyt Tidsskrift X B, 73–81 (1899)Google Scholar
  35. 213.
    Nielsen, N.: Note sur les développements schloemilchiens en série de fonctions cylindriques. Oversigt K. Danske Videnskabernes Selskabs 661–665 (1899)Google Scholar
  36. 214.
    Nielsen, N.: Sur le développement de zéro en fonctions cylindriques. Math. Ann. LII, 582–587 (1899)Google Scholar
  37. 215.
    Nielsen, N.: Note supplémentaire relative aux développements schloemilchiens en série de fonctions cylindriques. Oversigt K. Danske Videnskabernes Selskabs 55–60 (1900)Google Scholar
  38. 216.
    Nielsen, N.: Recherches sur une classe de séries infinies analogue á celle de M. W. Kapteyn. Oversigt K. Danske Videnskabernes Selskabs 127–146 (1901)Google Scholar
  39. 218.
    Nielsen, N.: Sur une classe de séries infinies analogues á celles de Schlömilch selon les fonctions cylindriques. Ann. di Mat. VI(3), 301–329 (1901)Google Scholar
  40. 219.
    Nielsen, N.: Handbuch der Theorie der Cylinderfenktionen. Teubner, Leipzig (1904)Google Scholar
  41. 245.
    Pogány, T.K., Parmar, R.K.: On p–extended Mathieu series. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. (2018) (to appear)Google Scholar
  42. 255.
    Popov, A.: On some summation formulas. Bull. Acad. Sci. LURSS 7, 801802 (1934). (in Russian)Google Scholar
  43. 256.
    Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series, vol. 1. Elementary Functions. Gordon and Breach Science Publishers, New York (1986)Google Scholar
  44. 257.
    Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series, vol. 2. Special Functions. Gordon and Breach Science Publishers, New York (1986)Google Scholar
  45. 258.
    Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series. Direct Laplace Transforms, vol. 4. Gordon and Breach Science Publishers, New York (1992)Google Scholar
  46. 264.
    Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988).Google Scholar
  47. 265.
    Rawn, M.D.: On the summation of Fourier and Bessel series. J. Math. Anal. Appl. 193, 282–295 (1995)Google Scholar
  48. 266.
    Rayleigh, J.W.S.: On a Physical Interpretation of Schlömilch’s Theorem in Bessel’s Functions. Philos. Mag. 6, 567–571 (1911)Google Scholar
  49. 273.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, New York (1993)Google Scholar
  50. 276.
    Schläfli, L.: Sopra un teorema di Jacobi recato a forma piu generale ed applicata alia funzione cilindrica. Ann. di Mat. (2) 5, 199–205 (1873)Google Scholar
  51. 278.
    Schlömilch, O.X.: Note sur la variation des constantes arbitraires d’une intégrale définie. J. Math. XXXIII, 268–280 (1846)Google Scholar
  52. 279.
    Schlömilch, O.X.: Über die Bessel’schen Function. Zeitschrift für Math. und Phys. II, 137–165 (1857)Google Scholar
  53. 285.
    Sousa, E.: How to approximate the fractional derivative of order 1 < α < 2. In: Podlubny, I., Vinagre Jara, B.M., Chen, Y.Q., Feliu Batlle, V., Tejado Balsera, I. (eds.) Proceedings of the 4th IFAC Workshop Fractional Differentiation and Its Applications, Art. No. FDA10–019 (2010)Google Scholar
  54. 293.
    Srivastava, H.M., Tomovski, ž.: Some problems and solutions involving Mathieu’s series and its generalizations. J. Inequal. Pure Appl. Math. 5(2) Art. 45, 1–13 (2004)Google Scholar
  55. 313.
    Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Clarendon Press, Oxford (1948)Google Scholar
  56. 315.
    Tošić, D.: Some series of a product of Bessel functions. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 678–715, 105–110 (1980)Google Scholar
  57. 316.
    Tričković, S.B., Stanković, M.S., Vidanović, M.V., Aleksić, V.N.: Integral transforms and summation of some Schlömilch series. In: Proceedings of the 5th International Symposium on Mathematical Analysis and its Applications (Niška Banja, Serbia). Mat. Vesnik 54, 211–218 (2002)Google Scholar
  58. 317.
    Twersky, V.: Elementary function representations of Schlömilch series. Arch. Ration. Mech. Anal. 8, 323332 (1961)Google Scholar
  59. 326.
    Walfisz, A.: Gitterpunkte in mehrdimensionalen Kugeln. Monografie Matematyczne, vol. 33, Państwowe Wydawnictwo Naukowe, Warsaw (1957)Google Scholar
  60. 333.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Árpád Baricz
    • 1
    • 2
  • Dragana Jankov Maširević
    • 3
  • Tibor K. Pogány
    • 1
    • 4
  1. 1.John von Neumann Faculty of Informatics, Institute of Applied MathematicsÓbuda UniversityBudapestHungary
  2. 2.Department of EconomicsBabeş–Bolyai UniversityCluj–NapocaRomania
  3. 3.Department of MathematicsJosip Juraj Strossmayer University of OsijekOsijekCroatia
  4. 4.Faculty of Maritime StudiesUniversity of RijekaRijekaCroatia

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