Abstract
It is known that multiple zeta values whose weight and depth are of opposite parity can be written in terms of multiple zeta values of lower depth. This theorem is called parity result. Multiple zeta values are special values of the multiple polylogarithms and the parity result is generalized to functional relations satisfied by the multiple polylogarithms. In this paper, we consider q- and elliptic generalizations of the parity result. As a main result of this paper, we establish parity result for functions \(L_{\varvec{k}}({\varvec{a}},{\varvec{\alpha }};p,q)\), which can be considered to be common deformations of q- and elliptic multiple polylogarithms. By taking the trigonometric and classical limits in the main theorem, we obtain q- and elliptic analogues of the parity result.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
References
Beilinson, A., Levin, A.: The elliptic polylogarithm. In : Motives, Proc. Sympos. Pure Math. American Mathematical Society, Providence, vol. 55, pp. 123–190 (1994)
Broedel, J., Kaderli, A.: Functional relations for elliptic polylogarithms. J. Phys. A 53(24), 245201 (2020)
Broedel, J., Duhr, C., Dulat, F., Tancredi, L.: Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism. J. High Energy Phys. JHEP 5, 1805 (2018)
Brown, F., Levin, A.: Multiple elliptic polylogarithms. arXiv:1110.6917
Guo, L., Xie, B.: Explicit double shuffle relations and a generalization of Euler’s decomposition formula. J. Algebra 380, 46–77 (2013)
Ihara, K., Kaneko, M., Zagier, D.: Derivation and double shuffle relations for multiple zeta values. Compos. Math. 142, 307–338 (2006)
Jonquière, A.: Note sure la série \(\sum _{n=1}^{\infty } \frac{x^n}{n^s}\). Bull. Soc. Math. France 17, 142–152 (1889)
Kato, M.: On certain two-parameter deformations of multiple zeta values. Res. Number Theory 6, 22 (2020)
Kato, M.: Sums of two-parameter deformations of multiple polylogarithms. Math. Phys. Anal. Geom 24, 23 (2021)
Kato, M.: On \(q\)-analogues of zeta functions of root systems. Kyushu J. Math 76(2), 451–475 (2022)
Kato, M., Takeyama, Y.: A deformation of multiple \(L\)-values. Ramanujan J. 57(1), 93–118 (2022)
Kusunoki, Y., Nakamura, Y., Sasaki, Y.: Analytic continuation of double polylogarithm by means of residue calculus. Comment. Math. Univ. St. Pauli 67, 49–64 (2019)
Kusunoki, Y., Nakamura, Y., Sasaki, Y.: A functional relation for analytic continuation of a multiple polylogarithm. Acta Arith. 195, 131–148 (2020)
Levin, A.: Elliptic polylogarithms: an analytic theory. Compos. Math. 106, 267–282 (1997)
Li, Z., Wakabayashi, N.: Sum of interpolated multiple \(q\)-zeta values. J. Number Theory 200, 205–259 (2019)
Nakamura, T.: Double Lerch value relations and functional relations for Witten zeta functions. Tokyo J. Math. 31, 551–574 (2008)
Panzer, E.: The parity theorem for multiple polylogarithms. J. Number Theory 172, 93–113 (2017)
Tsumura, H.: Combinatorial relations for Euler–Zagier sums. Acta Arith. 111(1), 27–42 (2004)
Zhao, J.: Multiple \(q\)-zeta functions and multiple \(q\)-polylogarityms. Ramanujan J. 14(2), 189–221 (2007)
Acknowledgements
The author would like to thank the referee for careful reading of the manuscript and constructive comments. He also would like to express his gratitude to Professor Yayoi Nakamura for her valuable comments. This work was supported by JSPS KAKENHI Grant Number JP20K14289.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kato, M. Parity result for q- and elliptic analogues of multiple polylogarithms. Res. number theory 9, 45 (2023). https://doi.org/10.1007/s40993-023-00452-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40993-023-00452-y