Tokyo Journal of Mathematics

New Trigonometric Identities and Reciprocity Laws of Generalized Dedekind Sums


Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this paper, we prove new trigonometric identities, which are product-to-sum type formulas for the higher derivatives of the cotangent and cosecant functions. Furthermore, from specializations of our formulas, we derive various known and new reciprocity laws of generalized Dedekind sums.

Article information

Tokyo J. Math., Volume 39, Number 2 (2016), 329-349.

First available in Project Euclid: 20 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11L03: Trigonometric and exponential sums, general
Secondary: 11F20: Dedekind eta function, Dedekind sums


SHIBUKAWA, Genki. New Trigonometric Identities and Reciprocity Laws of Generalized Dedekind Sums. Tokyo J. Math. 39 (2016), no. 2, 329--349. doi:10.3836/tjm/1484903126.

Export citation


  • M. Beck, Dedekind cotangent sums, Acta Arith. 109-2 (2003), 109–130.
  • U. Dieter, Cotangent sums, a further generalization of Dedekind sums, J. Number Theory 18-3 (1984), 289–305.
  • S. Egami, An elliptic analogue of the multiple Dedekind sums, Compositio Math. 99-1 (1995), 99–103.
  • S. Fukuhara, Dedekind symbols associated with J-forms and their reciprocity law, J. Number Theory 98-2 (2003), 236–253.
  • S. Fukuhara, New trigonometric identities and generalized Dedekind sums, Tokyo J. Math. 26-1 (2003), 1–14.
  • S. Fukuhara and N. Yui, Elliptic Apostol sums and their reciprocity laws, Trans. Amer. Math. Soc. 356-10 (2004), 4237–4254.
  • P. L. Walker, Elliptic functions, Wiley (1996).