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On generating functions of multiple zeta values and generalized hypergeometric functions

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Abstract

Generating functions for sums of certain multiple zeta values with fixed weight, depth and i-heights are discussed. The functions are systematically expressed in terms of generalized hypergeometric functions. The expressions reproduce several known formulas for multiple zeta values as applications.

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References

  1. Aoki T., Ohno Y.: Sum relations for multiple zeta values and connection formulas for the Gauss hypergeometric functions. Publ. Res. Inst. Math. Sci. 41, 329–337 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aoki T., Kombu Y., Ohno Y.: A generating function for sum of multiple zeta values and its applications. Proc. Am. Math. Soc. 136, 387–395 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arakawa T., Kaneko M.: Multiple zeta values, poly-Bernoulli numbers, and related zeta functions. Nagoya Math. J. 153, 191–216 (1999)

    MathSciNet  Google Scholar 

  4. Borwein J.M., Bailey D.H., Girgensohn R.: Experimentation in Mathematics. A. K. Peters, Wellesley (2004)

    MATH  Google Scholar 

  5. Bowman D., Bradley D.M.: Resolution of some open problems concerning multiple zeta evaluations of arbitrary depth. Compos. Math. 139, 85–100 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bradley D.M.: Multiple q-zeta values. J. Algebra 283, 752–798 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Euler, L.: Meditationes circa singulare serierum genus. Novi Commun. Acad. Sci. Petropol. 20, 140–186 (1775) Reprinted in Opera Omnia, ser. I, 15, 217–267, B. G. Teubner, Berlin (1927).

  8. Hoffman M.E.: Multiple harmonic series. Pac. J. Math. 152, 275–290 (1992)

    MATH  Google Scholar 

  9. Hoffman M.E.: The algebra of multiple harmonic series. J. Algebra 194, 477–495 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hoffman M.E., Ohno Y.: Relations of multiple zeta values and their algebraic expression. J. Algebra 262, 332–347 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ihara K., Kaneko M., Zagier D.: Derivation and double shuffle relations for multiple zeta values. Compos. Math. 142, 307–338 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Li Z.-h.: Sum of multiple zeta values of fixed weight, depth and i-height. Math. Z. 258, 133–142 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ohno Y.: A generalization of the duality and sum formulas on the multiple zeta values. J. Number Theory 74, 39–43 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ohno Y.: Sum relations for multiple zeta values, zeta functions, topology and quantum physics. Dev. Math. 14, 131–144 (2005)

    Article  MathSciNet  Google Scholar 

  15. Ohno Y., Zagier D.: Multiple zeta values of fixed weight, depth, and height. Indag. Math. 12, 483–487 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ohno Y., Wakabayashi N.: Cyclic sum of multiple zeta values. Acta Arithmetica 123, 289–295 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ohno Y., Okuda J.: On the sum formula for the q-analogue of non-strict multiple zeta values. Proc. Am. Math. Soc. 135, 3029–3037 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Okuda J., Takeyama Y.: On relations for multiple q-zeta values. Ramanujan J. 14, 379–387 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ohno Y., Zudilin W.: Zeta stars. Commun. Number Theory Phys. 2, 325–347 (2008)

    MATH  MathSciNet  Google Scholar 

  20. Takeyama Y.: A q-analogue of non-strict multiple zeta values and basic hypergeometric series. Proc. Am. Math. Soc. 137, 2997–3002 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Tanaka T., Wakabayashi N.: An algebraic proof of the cyclic sum formula for multiple zeta values. J. Algebra 323, 766–778 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  22. Vasil’ev, D.V.: Some formulas for Riemann zeta function at integer points, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 81–84 (1996). English transl., Moscow Univ. Math. Bull. 51, 41–43 (1996)

  23. Wakabayashi, N.: On generating functions for certain sums of multiple zeta values and a formula of S. Zlobin. Kôkyûroku Bessatsu (To appear)

  24. Zagier D.: Values of zeta functions and their applications, First European Congress of Mathematics, vol. II (Paris, 1992). Prog. Math. 120, 497–512 (1994)

    MathSciNet  Google Scholar 

  25. Zlobin, S.A.: Generating functions for values of a multiple zeta function, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 55–59 (2005). English transl., Moscow Univ. Math. Bull. 60 44–48 (2005)

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Correspondence to Noriko Wakabayashi.

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Aoki, T., Ohno, Y. & Wakabayashi, N. On generating functions of multiple zeta values and generalized hypergeometric functions. manuscripta math. 134, 139–155 (2011). https://doi.org/10.1007/s00229-010-0388-7

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  • DOI: https://doi.org/10.1007/s00229-010-0388-7

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