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

Essouabri, Driss ; Matsumoto, Kohji ; Tsumura, Hirofumi

Canadian journal of mathematics, Tome 63 (2011) no. 2, pp. 241-276

Résumé

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.

Détail

DOI : 10.4153/CJM-2010-085-1

Mots-clés : 11M41, 40B05, 11B39, Zeta-functions, holomorphic continuation, recurrence sequences, Fibonacci numbers, sum formulas

Bibliographie

[1] [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, 107–116. doi:10.4064/aa98-2-1 Google Scholar

[2] [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. 1–16. Google Scholar

[3] [3] Arakawa, T. and Kaneko, M., Multiple zeta values, poly-Bernoulli numbers, and related zeta functions. Nagoya Math. J. 153(1999), 189–209. Google Scholar

[4] [4] Arakawa, T. and Kaneko, M., On multiple L-values. J. Math. Soc. Japan 56(2004), no. 4, 967–991. doi:10.2969/jmsj/1190905444 Google Scholar

[5] [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, 85–100. doi:10.1023/B:COMP.0000005036.52387.da Google Scholar

[6] [6] Carlitz, L., Eulerian numbers and polynomials. Math. Mag. 32(1958/1959), 247–260. doi:10.2307/3029225 Google Scholar

[7] [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, 1373–1402. doi:10.1112/S0010437X06002235 Google Scholar

[8] [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, 1–16. doi:10.2969/jmsj/06010001 Google Scholar

[9] [9] Dupont, J. L., On polylogarithms. Nagoya Math. J. 114(1989), 1–20. Google Scholar

[10] [10] Ecalle, J., Multizetas, perinomal numbers, arithmetical dimorphy, and ARI/GARI. Ann. Fac. Sci. Toulouse Math. 13(2004), no. 4, 683–708. Google Scholar

[11] [11] Ecalle, J., ARI/GARI, la dimorphie et l’arithmétique des multizêtas: un premier bilan. J. Théor. Nombres Bordeaux 15(2003), no. 2, 411–478. Google Scholar

[12] [12] Enjalbert, J. Y., Minh, H. N., Analytic and combinatoric aspects of Hurwitz polyzêtas. J. Théor. Nombres Bordeaux 19(2007), no. 3, 595–640. Google Scholar

[13] [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. Google Scholar

[14] [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, 429–483. Google Scholar

[15] [15] Essouabri, D., Zeta functions associated to Pascal's triangle mod p. Japan. J. Math. 31(2005), no. 1, 157–174. Google Scholar

[16] [16] Essouabri, D., Mixed zeta functions and application to some lattice points problems. arXiv:math/0505558v2. Google Scholar

[17] [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), 217–267. Google Scholar

[18] [18] Frobenius, G., Über die Bernoullischen Zahlen und die Eulerschen Polynome. Preuss. Akad. Wiss. Sitzungsber. (1910), no. 2, 809–847. Google Scholar

[19] [19] Goncharov, A. B., Multiple polylogarithms and mixed Tate motives. arXiv:math/0103059v4. Google Scholar

[20] [20] Hoffman, M. E., Multiple harmonic series. Pacific J. Math. 152(1992), no. 2, 275–290. Google Scholar

[21] [21] Lewin, L., Polylogarithms and associated functions. North-Holland Publishing Co., New York-Amsterdam, 1981. Google Scholar

[22] [22] Lichtin, B., The asymptotics of a lattice point problem associated to a finite number of polynomials. Duke Math. J. 63(1991), no. 1, 139–192. doi:10.1215/S0012-7094-91-06307-6 Google Scholar

[23] [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. 417–440. Google Scholar

[24] [24] Matsumoto, K., Asymptotic expansions of double zeta-functions of Barnes, of Shintani, and Eisenstein series. Nagoya Math. J. 172(2003), 59–102. Google Scholar

[25] [25] Matsumoto, K., Analytic properties of multiple zeta-functions in several variables. In: Number Theory, Dev. Math., 15, Springer, New York, 2006, pp. 153–173. Google Scholar

[26] [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, 267–274. Google Scholar

[27] [27] Matsumoto, K. and Tsumura, H., Functional relations among certain double polylogarithms and their character analogues. Šiauliai Math. Semin. 3(11)(2008), 189–205. Google Scholar

[28] [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, 117–126. Google Scholar

[29] [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. 1–3, 87–116. doi:10.1016/S0012-365X(99)00124-7 Google Scholar

[30] [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. Google Scholar

[31] [31] Whittaker, E. T. and G. N., Watson, A course of modern analysis. 4th ed., Cambridge University Press, Cambridge, 1927. Google Scholar

[32] [32] Zagier, D., Values of zeta functions and their applications. In: First European Congress of Mathematics (Paris, 1992), Vol. II, Birkhäuser, Basel, 1994, pp. 497–512. Google Scholar

[33] [33] Zhao, J., Analytic continuation of multiple zeta functions. Proc. Amer. Math. Soc. 128(2000), no. 5, 1275–1283. doi:10.1090/S0002-9939-99-05398-8 Google Scholar