New trigonometric identities and reciprocity laws of generalized Dedekind sums

TL;DR

This paper introduces new trigonometric identities related to higher derivatives of cotangent and cosecant functions, leading to both known and novel reciprocity laws for generalized Dedekind sums.

Contribution

It presents new product-to-sum formulas for derivatives of cotangent and cosecant, and derives new reciprocity laws for generalized Dedekind sums from these identities.

Findings

New trigonometric identities for higher derivatives of cotangent and cosecant.

Derivation of known reciprocity laws of generalized Dedekind sums.

Discovery of new reciprocity laws for generalized Dedekind sums.

Abstract

We obtain new trigonometric identities, which are some product-to-sum type formulas for the higher derivative of the cotangent and cosecant functions. Further, from specializations of our formulas, we derive not only various known reciprocity laws of generalized Dedekind sums but also new reciprocity laws of generalized Dedekind sums.