Tokyo Journal of Mathematics

Zeta Regularized Product Expressions for Multiple Trigonometric Functions

Nobushige KUROKAWA and Masato WAKAYAMA

Full-text: Open access

Abstract

We introduce a multiple analogue of the gamma function which differs from the one defined by Barnes [B]. Using this function, we give expressions of the multiple sine and cosine functions in terms of zeta regularized products. The expression of the multiple sine function can be interpreted as a reflection formula of this new multiple analogue of the gamma function.

Article information

Source
Tokyo J. of Math. Volume 27, Number 2 (2004), 469-480.

Dates
First available in Project Euclid: 5 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1244208402

Digital Object Identifier
doi:10.3836/tjm/1244208402

Mathematical Reviews number (MathSciNet)
MR2107596

Zentralblatt MATH identifier
1065.11066

Subjects
Primary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Secondary: 11M06: $\zeta (s)$ and $L(s, \chi)$

Citation

KUROKAWA, Nobushige; WAKAYAMA, Masato. Zeta Regularized Product Expressions for Multiple Trigonometric Functions. Tokyo J. of Math. 27 (2004), no. 2, 469--480. doi:10.3836/tjm/1244208402. https://projecteuclid.org/euclid.tjm/1244208402.


Export citation

References

  • E. W. Barnes, On the theory of the multiple gamma function, Trans. Cambridge Philos. Soc. 19 (1904), 374–425.
  • Ch. Deninger, On the $\Gamma$-factors attached to motives, Invent. Math. 104 (1991), 245–261.
  • L. Euler, Exercitiones analyticae, Novi commentarii academiae scientiarum Petropolitanae 17 (1772), 173–204 [Opera Omnia I-15, 131–167].
  • O. Hölder, Ueber eine transcendente Function, Göttingen Nachrichten 1886 Nr. 16, 514–522.
  • M. Hirano, N. Kurokawa and M. Wakayama, Half zeta functions, J. Ramanujan Math. Soc. 18 (2003), 1–15.
  • G. Illies, Regularized products and determinants, Commun. Math. Phys. 220 (2001), 69–94.
  • K. Kimoto, N. Kurokawa, C. Sonoki and M. Wakayama, Zeta regularizations and $q$-analogue of ring sine functions, Kyushu Math. J. 57 (2003), 197–215.
  • K. Kimoto and M. Wakayama, Remarks on zeta regularized products. Intern. Math. Res. Notices 2004-17 (2004), 855–875.
  • N. Kurokawa and S. Koyama, Multiple sine functions, Forum Math. 15 (2003), 839–876.
  • N. Kurokawa, H. Ochiai and M. Wakayama, Multiple trigonometry and zeta functions, J. Ramanujan Math. Soc. 17 (2002), 101–113.
  • N. Kurokawa and M. Wakayama, On $\zeta(3)$, J. Ramanujan Math. Soc. 16 (2001), 205–214.
  • N. Kurokawa and M. Wakayama, Generalized zeta regularizations, quantum class number formulas, and Appell's $\mathcal{O}$-functions, to appear in The Ramanujan J.
  • N. Kurokawa and M. Wakayama, Extremal values of double and triple trigonometric functions, Kyushu J. Math. 58 (2004), 141–166.
  • N. Kurokawa and M. Wakayama, Finite ladders in multiple trigonometry, Preprint (2003).
  • M. Lerch, Dalši studie v oboru Malmsténovských řad, Rozpravy České Akad. 3 (1894), No. 28, 1–61.
  • E. C. Titchmarsh, The Theory of the Riemann Zeta-function, Oxford (1986), The 2nd edition.
  • A. Voros, Spectral functions, special functions and the Selberg zeta functions, Commun. Math. Phys. 110 (1987), 439–465.