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

Aoki, Takashi; Kombu, Yasuhiro; Ohno, Yasuo

doi:10.1090/S0002-9939-07-09175-7

A generating function for sums of multiple zeta values and its applications
HTML articles powered by AMS MathViewer

by Takashi Aoki, Yasuhiro Kombu and Yasuo Ohno PDF

Proc. Amer. Math. Soc. 136 (2008), 387-395 Request permission

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

Takashi Aoki and Yasuo Ohno, Sum relations for multiple zeta values and connection formulas for the Gauss hypergeometric functions, Publ. Res. Inst. Math. Sci. 41 (2005), no. 2, 329–337. MR 2138027
Tsuneo Arakawa and Masanobu Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J. 153 (1999), 189–209. MR 1684557, DOI 10.1017/S0027763000006954
J. M. Borwein, D. M. Bradley, and D. J. Broadhurst, Evaluations of $k$-fold Euler/Zagier sums: a compendium of results for arbitrary $k$, Electron. J. Combin. 4 (1997), no. 2, Research Paper 5, approx. 21. The Wilf Festschrift (Philadelphia, PA, 1996). MR 1444152
Karl Dilcher, Some $q$-series identities related to divisor functions, Discrete Math. 145 (1995), no. 1-3, 83–93. MR 1356587, DOI 10.1016/0012-365X(95)00092-B
A. Erdélyi et al. (eds.), Higher Transcendental Functions, vol. 1, Robert E. Krieger Publishing Company, Malabar, 1985.
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.
Andrew Granville, A decomposition of Riemann’s zeta-function, Analytic number theory (Kyoto, 1996) London Math. Soc. Lecture Note Ser., vol. 247, Cambridge Univ. Press, Cambridge, 1997, pp. 95–101. MR 1694987, DOI 10.1017/CBO9780511666179.009
Michael E. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992), no. 2, 275–290. MR 1141796
Michael E. Hoffman, Algebraic aspects of multiple zeta values, Zeta functions, topology and quantum physics, Dev. Math., vol. 14, Springer, New York, 2005, pp. 51–73. MR 2179272, DOI 10.1007/0-387-24981-8_{4}
Michael E. Hoffman and Yasuo Ohno, Relations of multiple zeta values and their algebraic expression, J. Algebra 262 (2003), no. 2, 332–347. MR 1971042, DOI 10.1016/S0021-8693(03)00016-4
Y. Kombu, Multiple zeta values and hypergeometric differential equations (in Japanese), Kinki University master’s thesis (2003).
Tu Quoc Thang Le and Jun Murakami, Kontsevich’s integral for the Homfly polynomial and relations between values of multiple zeta functions, Topology Appl. 62 (1995), no. 2, 193–206. MR 1320252, DOI 10.1016/0166-8641(94)00054-7
Yasuo Ohno, A generalization of the duality and sum formulas on the multiple zeta values, J. Number Theory 74 (1999), no. 1, 39–43. MR 1670544, DOI 10.1006/jnth.1998.2314
Yasuo Ohno, Sum relations for multiple zeta values, Zeta functions, topology and quantum physics, Dev. Math., vol. 14, Springer, New York, 2005, pp. 131–144. MR 2179276, DOI 10.1007/0-387-24981-8_{8}
Yasuo Ohno and Noriko Wakabayashi, Cyclic sum of multiple zeta values, Acta Arith. 123 (2006), no. 3, 289–295. MR 2263259, DOI 10.4064/aa123-3-5
Yasuo Ohno and Don Zagier, Multiple zeta values of fixed weight, depth, and height, Indag. Math. (N.S.) 12 (2001), no. 4, 483–487. MR 1908876, DOI 10.1016/S0019-3577(01)80037-9
Jun-ichi Okuda and Kimio Ueno, Relations for multiple zeta values and Mellin transforms of multiple polylogarithms, Publ. Res. Inst. Math. Sci. 40 (2004), no. 2, 537–564. MR 2049646
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
Don Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Paris, 1992) Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 497–512. MR 1341859
D. Zagier, Multiple zeta values. Unpublished manuscript, Bonn, 1995.