A $q$-analogue of non-strict multiple zeta values and basic hypergeometric series
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- by Yoshihiro Takeyama PDF
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Abstract:
We consider the generating function for a $q$-analogue of non-strict multiple zeta values (or multiple zeta-star values) and prove an explicit formula for it in terms of a basic hypergeometric series ${}_{3}\phi _{2}$. By specializing the variables in the generating function, we reproduce the sum formula obtained by Ohno and Okuda and get some relations in the case of full height.References
- Takashi Aoki, Yasuhiro Kombu, and Yasuo Ohno, A generating function for sums of multiple zeta values and its applications, Proc. Amer. Math. Soc. 136 (2008), no. 2, 387–395. MR 2358475, DOI 10.1090/S0002-9939-07-09175-7
- Takashi Aoki, Yasuo Ohno and Noriko Wakabayashi, Multiple zeta-star values with fixed weight, depth and $i$-heights and generalized hypergeometric functions, in preparation.
- David M. Bradley, Multiple $q$-zeta values, J. Algebra 283 (2005), no. 2, 752–798. MR 2111222, DOI 10.1016/j.jalgebra.2004.09.017
- David M. Bradley, Duality for finite multiple harmonic $q$-series, Discrete Math. 300 (2005), no. 1-3, 44–56. MR 2170113, DOI 10.1016/j.disc.2005.06.008
- George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR 2128719, DOI 10.1017/CBO9780511526251
- Zhong-hua Li, Sum of multiple zeta values of fixed weight, depth and $i$-height, Math. Z. 258 (2008), no. 1, 133–142. MR 2350039, DOI 10.1007/s00209-007-0163-y
- Yasuo Ohno and Jun-Ichi Okuda, On the sum formula for the $q$-analogue of non-strict multiple zeta values, Proc. Amer. Math. Soc. 135 (2007), no. 10, 3029–3037. MR 2322731, DOI 10.1090/S0002-9939-07-08994-0
- Jun-ichi Okuda and Yoshihiro Takeyama, On relations for the multiple $q$-zeta values, Ramanujan J. 14 (2007), no. 3, 379–387. MR 2357443, DOI 10.1007/s11139-007-9053-5
- 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
- Jianqiang Zhao, Multiple $q$-zeta functions and multiple $q$-polylogarithms, Ramanujan J. 14 (2007), no. 2, 189–221. MR 2341851, DOI 10.1007/s11139-007-9025-9
Additional Information
- Yoshihiro Takeyama
- Affiliation: Department of Mathematics, Graduate School of Pure and Applied Sciences, Tsukuba University, Tsukuba, Ibaraki 305-8571, Japan
- Email: takeyama@math.tsukuba.ac.jp
- Received by editor(s): August 18, 2008
- Published electronically: May 4, 2009
- Additional Notes: The research of the author was supported by Grant-in-Aid for Young Scientists (B) No. 20740088
- Communicated by: Peter A. Clarkson
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 2997-3002
- MSC (2000): Primary 33D15, 05A30, 11M41
- DOI: https://doi.org/10.1090/S0002-9939-09-09931-6
- MathSciNet review: 2506458