|
On the values of a class of Dirichlet series at rational arguments
Author(s):
K.
Chakraborty;
S.
Kanemitsu;
H.-L.
Li
Journal:
Proc. Amer. Math. Soc.
138
(2010),
1223-1230.
MSC (2010):
Primary 11M35;
Secondary 33B15
Posted:
December 4, 2009
MathSciNet review:
2578516
Retrieve article in:
PDF
Abstract |
References |
Similar articles |
Additional information
Abstract:
In this paper we shall prove that the combination of the general distribution property and the functional equation for the Lipschitz-Lerch transcendent capture the whole spectrum of deeper results on the relations between the values at rational arguments of functions of a class of zeta-functions. By Theorem 1 and its corollaries, we can cover all the previous results in a rather simple and lucid way. By considering the limiting cases, we can also deduce new striking identities for Milnor's gamma functions, among which is the Gauss second formula for the digamma function.
References:
-
- [Ap]
- T. M. Apostol, Introduction to analytic number theory, Springer-Verlag, New York-Heidelberg, 1976. MR 0434929 (55:7892)
- [CK1]
- D. Cvijović and J. Klinowski, New formula for the Bernoulli and Euler polynomials at rational arguments, Proc. Amer. Math. Soc. 123 (1995), 1527-1535. MR 1283544 (95g:11085)
- [CK2]
- D. Cvijović and J. Klinowski, Closed form summation of some trigonometric series, Math. Comput. 64 (1995), 205-210. MR 1270616 (95f:65017)
- [K]
- S. Kanemitsu, On evaluation of certain limits in closed form, Théorie des nombres, ed. by J.-M. De Koninck and C. Levesque, Walter de Gruyter, Berlin-New York, 1989, 459-474. MR 1024583 (90m:11127)
- [Vista]
- S. Kanemitsu and H. Tsukada, Vistas of special functions, World Scientific, Hackensack, NJ, 2007. MR 2352572
- [Le]
- L. Lewin, Polylogarithms and associated functions, North-Holland, New York, 1981. MR 618278 (83b:33019)
- [LHK]
- H.-L. Li, M. Hashimoto and S. Kanemitsu, On structural elucidation of Eisenstein's formula, to appear.
- [Mil]
- J. Milnor, On polylogarithms, Hurwitz zeta-functions and the Kubert identities, Enseign. Math. (2) 29 (1983), 281-322. MR 719313 (86d:11007)
- [MS]
- M. Ram Murty and N. Saradha, Transcendental values of the digamma function, J. Number Theory 125 (2007), 298-318. MR 2332591 (2008g:11123)
- [S1]
- H. M. Srivastava, A note on the closed-form summation of some trigonometric series, Kobe J. Math. 16 (1999), 177-182. MR 1745023 (2000m:11080)
- [S2]
- H. M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Camb. Philos. Soc. 129 (2000), 77-84. MR 1757780 (2001f:11033)
- [SC]
- H. M. Srivastava and J. Choi, Series associated with the zeta and related functions, Kluwer Academic Publishers, Dordrecht-Boston-London, 2001. MR 1849375 (2003a:11107)
- [Y]
- Y. Yamamoto, Dirichlet series with periodic coefficients, Proc. Intern. Sympos. ``Algebraic Number Theory'', Kyoto, 1976, Japan Soc. Promotional Sci., Tokyo, 1977, 275-289. MR 0450213 (56:8509)
Similar Articles:
Retrieve articles in Proceedings of the American Mathematical
Society
with
MSC (2010):
11M35,
33B15
Retrieve articles in all Journals with
MSC (2010):
11M35,
33B15
Additional Information:
K.
Chakraborty
Affiliation:
Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad 211 019, India
Email:
kalyan@hri.res.in
S.
Kanemitsu
Affiliation:
Graduate School of Advanced Technology, Kinki University, Iizuka, Fukuoka 820-8555, Japan
Email:
kanemitu@fuk.kindai.ac.jp
H.-L.
Li
Affiliation:
Department of Mathematics, Weinan Teachers College, Weinan, Shaanxi, 714000, People's Republic of China
Email:
lihailong@wntc.edu.cn
DOI:
10.1090/S0002-9939-09-10171-5
PII:
S 0002-9939(09)10171-5
Keywords:
Lipschitz-Lerch transcendent,
Hurwitz zeta function,
polylogarithm function,
Gauss' second formula,
Milnor's gamma function.
Received by editor(s):
December 28, 2008,
Received by editor(s) in revised form:
August 18, 2009
Posted:
December 4, 2009
Dedicated:
Dedicated to Professor Eiichi Bannai on his sixtieth birthday, with great respect and friendship
Communicated by:
Ken Ono
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
