Applications of the Hurwitz-Lerch Zeta-Function

Tomihiro Arai; Kalyan Chakraborty; Jing Ma

doi:doi:10.11648/j.pamj.s.2015040201.16

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Applications of the Hurwitz-Lerch Zeta-Function

Tomihiro Arai, Kalyan Chakraborty, Jing Ma

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

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

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[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.
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[17]	X.-H. Wang, Analytic continuation of the Riemann zeta-function, toappear.
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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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TY  - JOUR
T1  - Applications of the Hurwitz-Lerch Zeta-Function
AU  - Tomihiro Arai
AU  - Kalyan Chakraborty
AU  - Jing Ma
Y1  - 2014/12/27
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.s.2015040201.16
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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EP  - 35
PB  - Science Publishing Group
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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
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Author Information

Tomihiro Arai

Grad. School of Advances Tech. Kinki Univ.,Iizuka, Japan
Kalyan Chakraborty

Harish-Chandra Institute, Allahabad, Allahabad, India
Jing Ma

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

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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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - Applications of the Hurwitz-Lerch Zeta-Function
AU  - Tomihiro Arai
AU  - Kalyan Chakraborty
AU  - Jing Ma
Y1  - 2014/12/27
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.s.2015040201.16
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
SP  - 30
EP  - 35
PB  - Science Publishing Group
SN  - 2326-9812
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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