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Multiple Zeta-Functions Associated with Linear Recurrence Sequences and the Vectorial Sum Formula

  • Driss Essouabri (a1), Kohji Matsumoto (a2) and Hirofumi Tsumura (a3)

Abstract

We prove the holomorphic continuation of certain multi-variable multiple zeta-functions whose coefficients satisfy a suitable recurrence condition. In fact, we introduce more general vectorial zeta-functions and prove their holomorphic continuation. Moreover, we show a vectorial sum formula among those vectorial zeta-functions from which some generalizations of the classical sum formula can be deduced.

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References

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[1] Akiyama, S., Egami, S., and Tanigawa, Y., Analytic continuation of multiple zeta-functions and their values at non-positive integers. Acta Arith. 98(2001), no. 2, 107116. doi:10.4064/aa98-2-1
[2] Akiyama, S. and Ishikawa, H., On analytic continuation of multiple L-functions and related zeta-functions. In: Analytic number theory (Beijing/Kyoto, 1999), Dev. Math., 6, Kluwer, Dordrecht, 2002, pp. 116.
[3] Arakawa, T. and Kaneko, M., Multiple zeta values, poly-Bernoulli numbers, and related zeta functions. Nagoya Math. J. 153(1999), 189209.
[4] Arakawa, T. and Kaneko, M., On multiple L-values. J. Math. Soc. Japan 56(2004), no. 4, 967991. doi:10.2969/jmsj/1190905444
[5] Bowman, D. and Bradley, D. M., Resolution of some open problems concerning multiple zeta evaluations of arbitrary depth. Compositio Math. 139(2003), no. 1, 85100. doi:10.1023/B:COMP.0000005036.52387.da
[6] Carlitz, L., Eulerian numbers and polynomials. Math. Mag. 32(1958/1959), 247260. doi:10.2307/3029225
[7] de Crisenoy, M., Values at T-tuples of negative integers of twisted multivariable zeta series associated to polynomials of several variables. Compos. Math. 142(2006), no. 6, 13731402. doi:10.1112/S0010437X06002235
[8] de Crisenoy, M. and Essouabri, D., Relations between values at T-tuples of negative integers of twisted multi-variable zeta series associated to polynomials of several variables. J. Math. Soc. Japan 60(2008), no. 1, 116. doi:10.2969/jmsj/06010001
[9] Dupont, J. L., On polylogarithms. Nagoya Math. J. 114(1989), 120.
[10] Ecalle, J., Multizetas, perinomal numbers, arithmetical dimorphy, and ARI/GARI. Ann. Fac. Sci. Toulouse Math. 13(2004), no. 4, 683708.
[11] Ecalle, J., ARI/GARI, la dimorphie et l’arithmétique des multizêtas: un premier bilan. J. Théor. Nombres Bordeaux 15(2003), no. 2, 411478.
[12] Enjalbert, J. Y., Minh, H. N., Analytic and combinatoric aspects of Hurwitz polyzêtas. J. Théor. Nombres Bordeaux 19(2007), no. 3, 595640.
[13] Essouabri, D., Singularités des séries de Dirichlet associées à des polynômes de plesieurs variables et applications à la théorie analytique des nombres. Thèse, Univ. Nancy I, 1995.
[14] Essouabri, D., Singularités des séries de Dirichlet associées à des polynômes de plusieurs variables et applications en théorie analytique des nombres. Ann. Inst. Fourier (Grenoble) 47(1997), no. 2, 429483.
[15] Essouabri, D., Zeta functions associated to Pascal's triangle mod p. Japan. J. Math. 31(2005), no. 1, 157174.
[16] Essouabri, D., Mixed zeta functions and application to some lattice points problems. arXiv:math/0505558v2.
[17] Euler, L., Meditationes circa singulare serierum genus. Novi Comm. Acad. Sci. Petropol. 20(1775), 140–186; reprinted in Opera Omnia, ser. I 15(1927), 217267.
[18] Frobenius, G., Über die Bernoullischen Zahlen und die Eulerschen Polynome. Preuss. Akad. Wiss. Sitzungsber. (1910), no. 2, 809847.
[19] Goncharov, A. B., Multiple polylogarithms and mixed Tate motives. arXiv:math/0103059v4.
[20] Hoffman, M. E., Multiple harmonic series. Pacific J. Math. 152(1992), no. 2, 275290.
[21] Lewin, L., Polylogarithms and associated functions. North-Holland Publishing Co., New York-Amsterdam, 1981.
[22] Lichtin, B., The asymptotics of a lattice point problem associated to a finite number of polynomials. Duke Math. J. 63(1991), no. 1, 139192. doi:10.1215/S0012-7094-91-06307-6
[23] Matsumoto, K., On the analytic continuation of various multiple zeta-functions. In: Number theory for the millennium, II (Urbana IL, 2000), A K Peters, Natick, MA, 2002, pp. 417440.
[24] Matsumoto, K., Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series. Nagoya Math. J. 172(2003), 59102.
[25] Matsumoto, K., Analytic properties of multiple zeta-functions in several variables. In: Number Theory, Dev. Math., 15, Springer, New York, 2006, pp. 153173.
[26] Matsumoto, K. and Tanigawa, Y., The analytic continuation and the order estimate of multiple Dirichlet series. J. Théor. Nombres Bordeaux 15(2003), no. 1, 267274.
[27] Matsumoto, K. and Tsumura, H., Functional relations among certain double polylogarithms and their character analogues. Šiauliai Math. Semin. 3(11)(2008), 189205.
[28] Minh, H. N., Algebraic combinatoric aspects of asymptotic analysis of nonlinear dynamical system with singular inputs. In: CADE-2007, Computer Algebra and Differential Equations, Acta Academiae Aboensis, Ser. B, 67(2007), no. 2, 117126.
[29] Minh, H. N. and Jacob, G., Symbolic integration of meromorphic differential systems via Dirichlet functions. Formal power series and algebraic combinatorics (Minneapolis, MN, 1996). Discrete Math. 210(2000), no. 13, 87116. doi:10.1016/S0012-365X(99)00124-7
[30] Philippou, A. N., Horadam, A. F., and Bergum, G. E. (eds), Applications of Fibonacci numbers. Proceedings of the Second International Conference on Fibonacci Numbers and their Applications held at San Jose State University, San Jose, California, August 13–16, 1986. Kluwer Academic Publishers, Dordrecht, 1988.
[31] Whittaker, E. T. and G. N., Watson, A course of modern analysis. 4th ed., Cambridge University Press, Cambridge, 1927.
[32] Zagier, D., Values of zeta functions and their applications. In: First European Congress of Mathematics (Paris, 1992), Vol. II, Birkhäuser, Basel, 1994, pp. 497512.
[33] Zhao, J., Analytic continuation of multiple zeta functions. Proc. Amer. Math. Soc. 128(2000), no. 5, 12751283. doi:10.1090/S0002-9939-99-05398-8
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Multiple Zeta-Functions Associated with Linear Recurrence Sequences and the Vectorial Sum Formula

  • Driss Essouabri (a1), Kohji Matsumoto (a2) and Hirofumi Tsumura (a3)

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