Checkerboard style Schur multiple zeta values and odd single zeta values

Abstract

We give explicit formulas for the recently introduced Schur multiple zeta values, which generalize multiple zeta(-star) values and which assign to a Young tableaux a real number. In this note we consider Young tableaux of various shapes, filled with alternating entries like a Checkerboard. In particular we obtain new sum representation for odd single zeta values in terms of these Schur multiple zeta values. As a special case we show that some Schur multiple zeta values of Checkerboard style, filled with 1 and 3, are given by determinants of matrices with odd single zeta values as entries.

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Acknowledgements

The authors would like to thank Don Zagier for his ideas on parts of the proof of Theorem 3.5, Wadim Zudilin for helpful discussion on the topic, Steven Charlton for corrections and calculations in Section 5 and the Max-Planck-Institut für Mathematik in Bonn for hospitality and support.

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Correspondence to Henrik Bachmann.

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Y. Yamasaki is partially supported by JSPS Grant-in-Aid for Scientific Research (C) No. 15K04785.

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Bachmann, H., Yamasaki, Y. Checkerboard style Schur multiple zeta values and odd single zeta values. Math. Z. 290, 1173–1197 (2018). https://doi.org/10.1007/s00209-018-2058-5

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Keywords

  • Multiple zeta values
  • Schur functions
  • Jacobi–Trudi formula
  • Hypergeometric functions
  • Hankel determinants

Mathematics Subject Classification

  • 11M41
  • 05E05
  • 33C05