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Applications of the Hurwitz-Lerch Zeta-Function

Received: 5 November 2014     Accepted: 14 November 2014     Published: 27 December 2014
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Abstract

In this paper, we shall exhibit the use of two principles, “principle of decomposition into residue classes” and “binomial principle of analytic continuation” due to Ram Murty and Sinha and indicate a certain distribution property and the functional equation for the Lipschitz-Lerch transcendent at integral arguments ofs. By considering the limiting cases ,we can also deduce new striking identities for Lipschizt-Lerch transcendent, among which is the Gauss second formula for the digamma function, Lipschitz-Lerch transcendent

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 2-1)

This article belongs to the Special Issue Abridging over Troubled Water---Scientific Foundation of Engineering Subjects

DOI 10.11648/j.pamj.s.2015040201.16
Page(s) 30-35
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

Decomposition into Residue Classes, Binomial Expansion, Distribution Property, Zeta-Functions, Functional Equation

References
[1] K. Chakraborty, S. Kanemitsu, and H. -L. Li, On the values of a classof Dirichlet series at rational arguments, Proc. Amer. Math. Soc. 138(2010), 1223-1230.
[2] G. Eisenstein, Aufgaben und Lehrs¨atze, J. Reine Angew. Math. 27 (1844), 281-283=Math. Werke, Vol. 1, 1975, Chelsea, 108-110.
[3] A. Erdléyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Highertrancendental functions vol 1, McGraw-hill, New York 1953.
[4] H. Hasse, EinSummierungsverfahrenf¨ur die Riemannsche ζ-Reihe,Math. Z. 32(1930), 458-464.
[5] A. Hurwitz, MathematischeWerke, Vol. 2, Basel, Birkh¨auser, 1932.
[6] M. Ishibashi, An elementary proof of the generalized Eisenstein formula, Sitzungsber. Osterreich. Wiss. Wien, Math. -naturwiss. Kl. 197 (1988), 443-447.
[7] S. Kanemitsu, M. Katsurada, and M. Yoshimoto, On the Hurwitz-Lerchzeta function, Aequationes Math. 59 (2000), 1-19.
[8] S. Kanemitsu and H. Tsukada, Vistas of special functions, World Scientific,Singapore-London-New York. 2007.
[9] D. Klusch, On the Taylor expansion for the Lerch zeta function, J. Math.Anal. Appl. 170 (1992), 513-523.
[10] H. -L. Li, M. Hashimoto and S. Kanemitsu, On structural elucidationof Eisenstein’s formula, Sci. China. 53 (2010), 2341-2350.
[11] T. Nakamura, Some topics related to Hurwitz-Lerch zeta functions, TheRamanujan J. 21 (2010), 285-302.
[12] M. Ram Murty and Kaneenika Sinha, Multiple Hurwitz zeta functions,Proc. Sympos. Pure Math.75 (2006), 135-156.
[13] B. Riemann, ¨Uber die Anzahl der Primzahlen, untereinergegebenenGr¨osse, Monatsber. Akad. Berlin, (1859), 1-680 =Collected Works ofBernhard Riemann, ed. by H. Weber, 2nd ed. Dover, New York 1953.
[14] J. Sondow, Analytic continuation of Riemann’s zeta function and valuesat negative integers via Euler’s transformation of series, Proc. Amer.Math. Soc. 120 (1994), 421-424.
[15] H. M. Srivastava and J. -S. Choi, Series associated with the Zeta and relatedfunctions, Kluwer Academic Publishers, Dordrecht-Boston-London2001.
[16] H. M. Stark, Dirichlet’s class-number formula revisited, Contemp. Math.143 (1993), 571-577.
[17] X.-H. Wang, Analytic continuation of the Riemann zeta-function, toappear.
[18] J. R. Wilton, A proof of Burnside’s formula for and certainallied properties of Riemann’s ζ-function, Mess. Math.55 (1922/1923),90-93.
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  • APA Style

    Tomihiro Arai, Kalyan Chakraborty, Jing Ma. (2014). Applications of the Hurwitz-Lerch Zeta-Function. Pure and Applied Mathematics Journal, 4(2-1), 30-35. https://doi.org/10.11648/j.pamj.s.2015040201.16

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    ACS Style

    Tomihiro Arai; Kalyan Chakraborty; Jing Ma. Applications of the Hurwitz-Lerch Zeta-Function. Pure Appl. Math. J. 2014, 4(2-1), 30-35. doi: 10.11648/j.pamj.s.2015040201.16

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    AMA Style

    Tomihiro Arai, Kalyan Chakraborty, Jing Ma. Applications of the Hurwitz-Lerch Zeta-Function. Pure Appl Math J. 2014;4(2-1):30-35. doi: 10.11648/j.pamj.s.2015040201.16

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  • @article{10.11648/j.pamj.s.2015040201.16,
      author = {Tomihiro Arai and Kalyan Chakraborty and Jing Ma},
      title = {Applications of the Hurwitz-Lerch Zeta-Function},
      journal = {Pure and Applied Mathematics Journal},
      volume = {4},
      number = {2-1},
      pages = {30-35},
      doi = {10.11648/j.pamj.s.2015040201.16},
      url = {https://doi.org/10.11648/j.pamj.s.2015040201.16},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2015040201.16},
      abstract = {In this paper, we shall exhibit the use of two principles, “principle of decomposition into residue classes” and “binomial principle of analytic continuation” due to Ram Murty and Sinha and indicate a certain distribution property and the functional equation for the Lipschitz-Lerch transcendent at integral arguments ofs. By considering the limiting cases ,we can also deduce new striking identities for Lipschizt-Lerch transcendent, among which is the Gauss second formula for the digamma function, Lipschitz-Lerch transcendent},
     year = {2014}
    }
    

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    T1  - Applications of the Hurwitz-Lerch Zeta-Function
    AU  - Tomihiro Arai
    AU  - Kalyan Chakraborty
    AU  - Jing Ma
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    DO  - 10.11648/j.pamj.s.2015040201.16
    T2  - Pure and Applied Mathematics Journal
    JF  - Pure and Applied Mathematics Journal
    JO  - Pure and Applied Mathematics Journal
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    UR  - https://doi.org/10.11648/j.pamj.s.2015040201.16
    AB  - In this paper, we shall exhibit the use of two principles, “principle of decomposition into residue classes” and “binomial principle of analytic continuation” due to Ram Murty and Sinha and indicate a certain distribution property and the functional equation for the Lipschitz-Lerch transcendent at integral arguments ofs. By considering the limiting cases ,we can also deduce new striking identities for Lipschizt-Lerch transcendent, among which is the Gauss second formula for the digamma function, Lipschitz-Lerch transcendent
    VL  - 4
    IS  - 2-1
    ER  - 

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Author Information
  • Grad. School of Advances Tech. Kinki Univ.,Iizuka, Japan

  • Harish-Chandra Institute, Allahabad, Allahabad, India

  • Dept. of Math., Jilin Univ., Changchun, PRC

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