Skip to main content
SpringerLink
Log in

p-Extended Mathieu Series from the Schlömilch Series Point of View

  • Published:
Vietnam Journal of Mathematics Aims and scope Submit manuscript

Abstract

Motivated by certain current results by Pogány and Parmar [10] in which the authors introduced the so-called p-extended Mathieu series, the main aim of this paper is to present a connection between such series and various types of Schlömilch series.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Actually, A p, q (γ) is the Laplace transform of x↦x −1 e −p/x J ν (γx) at the argument q.

References

  1. Cerone, P., Lenard, C.T.: On integral forms of generalized mathieu series. J. Inequal. Pure Appl. Math. 4(100), 11 (2003)

    MATH  Google Scholar 

  2. Chaudhry, M.A., Qadir, A., Boudjelkha, M.T., Rafique, M., Zubair, S.M.: Extended Riemann zeta functions. Rocky Mt. J. Math. 31, 1237–1263 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products. In: Jeffrey, A., Zwillinger, D. (eds.) . 6th edn. Academic, New York (2000)

  5. Jankov, D., Pogány, T.K.: Integral representation of Schlömilch series. J. Class. Anal. 1, 75–84 (2012)

    Article  MathSciNet  Google Scholar 

  6. Mathieu, E.́L.: Traité De Physique Mathématique, VI-VII: Théorie De L’élasticitré Des Corps Solides. Gauthier–Villars, Paris (1890)

  7. Milovanović, G.V., Pogány, T.K.: New integral forms of generalized Mathieu series and related applications. Appl. Anal. Discrete Math. 7, 180–192 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mortici, C.: Accurate approximations of the Mathieu series. Math. Comput. Model. 53, 909–914 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Murio, D.A.: Stable numerical evaluation of grünwald–letnikov fractional derivatives applied to a fractional IHCP. Inverse Probl. Sci. Eng. 17, 229–243 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pogány, T.K., Parmar, R.K. On p-extended Mathieu series. arXiv:1606.08369 (2016)

  11. Pogány, T.K., Srivastava, H.M., Tomovski, Z.: Some families of Mathieu a-series and alternating Mathieu a-series. Appl. Math. Comput. 173, 69–108 (2006)

    MathSciNet  MATH  Google Scholar 

  12. Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series, vol. 2. Special Functions. Gordon and Breach Science Publishers, New York (1986)

  13. Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series, vol. 4. Direct Laplace Transforms. Gordon and Breach Science Publishers, New York (1992)

  14. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Yverdon (1993)

    MATH  Google Scholar 

  15. 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)

  16. Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)

    MATH  Google Scholar 

Download references

Acknowledgments

Both authors are grateful to the anonymous referees for the comments which significantly contributed to improve the quality of the exposition.

This article has been partially supported by Hrvatska Zaklada za Znanost (HR) grant No. HRZZ-5435.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Osijek, Trg Lj. Gaja 6, 31000, Osijek, Croatia

    Dragana Jankov Maširević

  2. Faculty of Maritime Studies, University of Rijeka, Studentska 2, 51000, Rijeka, Croatia

    Tibor K. Pogány

  3. Institute of Applied Mathematics, Óbuda University, 1034, Budapest, Hungary

    Tibor K. Pogány

Authors
  1. Dragana Jankov Maširević
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tibor K. Pogány
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tibor K. Pogány.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Maširević, D.J., Pogány, T.K. p-Extended Mathieu Series from the Schlömilch Series Point of View. Vietnam J. Math. 45, 713–719 (2017). https://doi.org/10.1007/s10013-017-0241-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10013-017-0241-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation