We present explicit formulas for Hecke eigenforms as linear combinations of q-analogues of modified double zeta values. As an application, we obtain period polynomial relations and sum formulas for these modified double zeta values. These relations have similar shapes as the period polynomial relations of Gangl, Kaneko, and Zagier and the usual sum formulas for classical double zeta values.
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Bachmann, H., Kühn, U.: The algebra of generating functions for multiple divisor sums and applications to multiple zeta values. Ramanujan J. 40, 605–648 (2016)
Borwein, J., Bradley, D., Broadhurst, D.: Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k. Electron. J. Combin. 4(2), 1–21 (1997)
Choie, Y., Zagier, D.: Rational period functions for PSL(2, Z). Contemp. Math. 143, 89–108 (1993)
Gangl, H., Kaneko, M., Zagier, D.: Automorphic forms and zeta functions. Double zeta values and modular forms. Automorphic forms and zeta functions, pp. 71–106. World Science Publication, Hackensack, NJ (2006)
Kohnen, W., Zagier, D.: Modular forms with rational periods. Modular forms, Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, pp. 197–249 (1984)
Ma, D., Tasaka, K.: Relationship between multiple zeta values of depths 2 and 3 and period polynomials, preprint, arXiv:1707.08178
Manin, Y.: Periods of parabolic forms and \(p\)-adic Hecke series. Mat. Sb. 21, 371–393 (1973)
Matsumoto, K.: On Mordell–Tornheim and other multiple zeta-functions. In Proceedings of the session in analytic number theory and diophantine equations. Bonner Math. Schriften 360 (2003)
Kaneko, M., Tasaka, K.: Double zeta values, double Eisenstein series, and modular forms of level 2. Math. Ann. 357(3), 1091–1118 (2013)
Okamoto, T.: Some relations among Apostol–Vu double zeta values for coordinate wise limits at non-positive integers. Tokyo J. Math. 34(2), 353–366 (2011)
Ohno, Y., Zudilin, W.: Zeta stars. Commun. Number Theory Phys. 2(2), 325–347 (2008)
Tasaka, K.: Hecke Eigenform and double Eisenstein series. Proc. Am. Math. Soc. 148(1), 53–58 (2020)
Wan, J.: Some notes on weighted sum formulae for double zeta values. Proc. Math. Stat. 43, 361–380 (2013)
Zagier, D.: Hecke operators and periods of modular forms. Israel Math. Conf. Proc. 3, 321–336 (1990)
Zagier, D.: Periods of modular forms and Jacobi theta functions. Invent. Math. 104(3), 449–465 (1991)
Zagier, D.: Periods of modular forms, traces of Hecke operators, and multiple zeta values. RIMS Kokyuroku 843, 162–170 (1993)
The author would like to thank Ulf Kühn, Nils Matthes and the referee for a lot fruitful comments and corrections.
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Communicated by Jens Funke.
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Bachmann, H. Modular forms and q-analogues of modified double zeta values. Abh. Math. Semin. Univ. Hambg. 90, 201–213 (2020). https://doi.org/10.1007/s12188-020-00227-7
- Modular forms
- Double zeta values
- Period polynomials
- Hecke operators
Mathematics Subject Classification
- Primary 11F11
- Secondary 11F67