Closed-form formulae for the derivatives of trigonometric functions at rational multiples of $\pi$

TL;DR

This paper derives closed-form expressions for all derivatives of four key trigonometric functions at rational multiples of pi, using special functions and finite sums involving Bernoulli and Euler polynomials.

Contribution

It provides a unified method to express derivatives of trigonometric functions at rational multiples of pi in closed form, expanding on previous isolated results.

Findings

Derivatives expressed as finite sums with Bernoulli and Euler polynomials.

Unified approach using special functions like Hurwitz and Lerch zeta functions.

Includes specific cases and applications of the formulas.

Abstract

In this sequel to our recent note it is shown, in a unified manner, by making use of some basic properties of certain special functions, such as the Hurwitz zeta function, Lerch zeta function and Legendre chi function, that the values of all derivatives of four trigonometric functions at rational multiples of $\pi$ can be expressed in closed form as simple finite sums involving the Bernoulli and Euler polynomials. In addition, some particular cases are considered.