Derivatives of tangent function and tangent numbers

Abstract

In the paper, by induction, the Faà 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.

17 pages

References in corpus (18)

• Eight interesting identities involving the exponential function, derivatives, and Stirling numbers of the second kind
• Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind
• Explicit expressions for a family of Bell polynomials and derivatives of some functions
• Limit formulas for ratios of polygamma functions at their singularities
• Limit formulas for ratios of derivatives of the gamma and digamma functions at their singularities
• Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind
• Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions
• Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions
• An integral representation and properties of Bernoulli numbers of the second kind
• An integral representation, some inequalities, and complete monotonicity of Bernoulli numbers of the second kind
• An explicit formula for Bernoulli numbers in terms of Stirling numbers of the second kind
• An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions
• An explicit formula for computing Bell numbers in terms of Lah and Stirling numbers
• An integral representation, complete monotonicity, and inequalities of Cauchy numbers of the second kind
• Properties of three functions relating to the exponential function and the existence of partitions of unity
• Alternative proofs of a formula for Bernoulli numbers in terms of Stirling numbers
• Some integral representations and properties of Lah numbers
• A new explicit formula for Bernoulli and Genocchi numbers in terms of Stirling numbers

Cited by in corpus (10)

• Two closed forms for the Bernoulli polynomials
• An explicit formula for Bernoulli polynomials in terms of $r$-Stirling numbers of the second kind
• Maclaurin's series expansions for positive integer powers of inverse (hyperbolic) sine and related functions, specific values of partial Bell polynomials, and two applications
• Taylor's series expansions for real powers of functions containing squares of inverse (hyperbolic) cosine functions, explicit formulas for special partial Bell polynomials, and series representations for powers of circular constant
• Several series expansions for real powers and several formulas for partial Bell polynomials of sinc and sinhc functions in terms of central factorial and Stirling numbers of second kind
• Closed formulas and determinantal expressions for higher-order Bernoulli and Euler polynomials in terms of Stirling numbers
• A determinantal expression and a recursive relation of the Delannoy numbers
• Some classes of generating functions for generalized Hermite- and Chebyshev-type polynomials: Analysis of Euler's formula
• Solution of the equation $y^{\prime }=f(y)$ and Bell Polynomials
• On the derivatives of the powers of trigonometric and hyperbolic sine and cosine