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A generating function for sums of multiple zeta values and its applications


Authors: Takashi Aoki, Yasuhiro Kombu and Yasuo Ohno
Journal: Proc. Amer. Math. Soc. 136 (2008), 387-395
MSC (2000): Primary 11M06, 40B05; Secondary 33C05
DOI: https://doi.org/10.1090/S0002-9939-07-09175-7
Published electronically: November 1, 2007
MathSciNet review: 2358475
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Abstract: A generating function for specified sums of multiple zeta values is defined and a differential equation that characterizes this function is given. As applications, some relations for multiple zeta values over the field of rational numbers are discussed.


References [Enhancements On Off] (What's this?)

  • 1. T. Aoki and Y. Ohno, Sum relations for multiple zeta values and connection formulas for the Gauss hypergeometric function, Publ. RIMS, Kyoto Univ. 41 (2005), 329-337. MR 2138027 (2005m:11165)
  • 2. T. Arakawa and M. Kaneko, Multiple zeta-values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J. 153 (1999), 1-21. MR 1684557 (2000e:11113)
  • 3. J. Borwein, D. Bradley and D. Broadhurst, Evaluations of $ k$-fold Euler/Zagier sums: a compendium of results for arbitrary $ k$. The Wilf Festschrift Volume. Electron. J. Combin. 4 (1997), no. 2, Research Paper 5, 19 pp. MR 1444152 (98b:11091)
  • 4. K. Dilcher, Some $ q$-series identities related to divisor functions, Disc. Math. 145 (1995), 83-93. MR 1356587 (96i:11020)
  • 5. A. Erdélyi et al. (eds.), Higher Transcendental Functions, vol. 1, Robert E. Krieger Publishing Company, Malabar, 1985.
  • 6. L. Euler, Meditationes circa singulare serierum genus, Novi Comm. Acad. Sci. Petropol 20 (1775), 140-186, reprinted in Opera Omnia ser. I, vol. 15, B. G. Teubner, Berlin (1927), 217-267.
  • 7. A. Granville, A decomposition of Riemann's zeta-function, in Analytic Number Theory, London Mathematical Society Lecture Note Series 247, Y. Motohashi (ed.), Cambridge University Press, (1997), 95-101. MR 1694987 (2000c:11134)
  • 8. M. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992), 275-290. MR 1141796 (92i:11089)
  • 9. M. Hoffman, Algebraic aspects of multiple zeta values, In Zeta Functions, Topology and Quantum Physics (T. Aoki, S. Kanemitsu, M. Nakahara and Y. Ohno, eds.), Developments in Mathematics 14, Springer, 2005, pp. 51-74. MR 2179272 (2006g:11185)
  • 10. M. Hoffman and Y. Ohno, Relations of multiple zeta values and their algebraic expression, J. Algebra 262 (2003), 332-347. MR 1971042 (2004c:11163)
  • 11. Y. Kombu, Multiple zeta values and hypergeometric differential equations (in Japanese), Kinki University master's thesis (2003).
  • 12. T. Q. T. Le and J. Murakami, Kontsevich's integral for the Homfly polynomial and relations between values of multiple zeta functions, Topology and its Applications 62 (1995), 193-206. MR 1320252 (96c:57017)
  • 13. Y. Ohno, A generalization of the duality and sum formulas on the multiple zeta values. J. Number Theory 74 (1999), 39-43. MR 1670544 (99k:11138)
  • 14. Y. Ohno, Sum relations for multiple zeta values. In Zeta Functions, Topology and Quantum Physics (T. Aoki, S. Kanemitsu, M. Nakahara and Y. Ohno eds.), Developments in Mathematics 14, Springer, 2005, pp. 131-144. MR 2179276 (2006i:11105)
  • 15. Y. Ohno and N. Wakabayashi, Cyclic sum of multiple zeta values, Acta Arithmetica 123 (2006), 289-295. MR 2263259 (2007g:11113)
  • 16. Y. Ohno and D. Zagier, Multiple zeta values of fixed weight, depth, and height, Indag. Math. 12 (2001), 483-487. MR 1908876 (2003e:11094)
  • 17. J. Okuda and K. Ueno, Relations for multiple zeta values and Mellin transforms of multiple polylogarithms. Publ. RIMS, Kyoto Univ. 40 (2004), 537-564. MR 2049646 (2005f:11199)
  • 18. E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd edition revised by D. R. Heath-Brown, Oxford University Press, Oxford, 1986. MR 882550 (88c:11049)
  • 19. D. Zagier, Values of zeta functions and their applications. In Proceedings of ECM 1992, Progress in Math. 120 (1994), 497-512. MR 1341859 (96k:11110)
  • 20. D. Zagier, Multiple zeta values. Unpublished manuscript, Bonn, 1995.

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Additional Information

Takashi Aoki
Affiliation: Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan
Email: aoki@math.kindai.ac.jp

Yasuhiro Kombu
Affiliation: Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan
Email: kombu@math.kindai.ac.jp

Yasuo Ohno
Affiliation: Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan
Email: ohno@math.kindai.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-07-09175-7
Keywords: Multiple zeta values, hypergeometric functions
Received by editor(s): August 2, 2006
Published electronically: November 1, 2007
Additional Notes: The first author was supported in part by JSPS Grant-in-Aid No. 18540197.
The third author was supported in part by JSPS Grant-in-Aid No. 18540197 and No. 18740020.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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