Derivatives of tangent function and tangent numbers

TL;DR

This paper derives explicit formulas for higher derivatives of tangent, cotangent, sine, and cosine functions, explores related special numbers and polynomials, and presents identities and recurrence relations for these mathematical entities.

Contribution

It introduces new explicit formulas, identities, and recurrence relations for derivatives of trigonometric functions and related special numbers, expanding existing mathematical knowledge.

Findings

Explicit formulas for derivatives of tangent and cotangent functions.

Recurrence relations for tangent, Bernoulli, and Genocchi numbers.

Identities involving sine, cosine, and special values of zeta and Euler polynomials.

Abstract

In the paper, by induction, the Fa\`a di Bruno formula, and some techniques in the theory of complex functions, the author finds explicit formulas for higher order derivatives of the tangent and cotangent functions as well as powers of the sine and cosine functions, obtains explicit formulas for two Bell polynomials of the second kind for successive derivatives of sine and cosine functions, presents curious identities for the sine function, discovers explicit formulas and recurrence relations for the tangent numbers, the Bernoulli numbers, the Genocchi numbers, special values of the Euler polynomials at zero, and special values of the Riemann zeta function at even numbers, and comments on five different forms of higher order derivatives for the tangent function and on derivative polynomials of the tangent, cotangent, secant, cosecant, hyperbolic tangent, and hyperbolic cotangent functions.