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Derivatives of tangent function and tangent numbers

Author

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  • Qi, Feng

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.

Suggested Citation

  • Qi, Feng, 2015. "Derivatives of tangent function and tangent numbers," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 844-858.
    • Handle: RePEc:eee:apmaco:v:268:y:2015:i:c:p:844-858
    DOI: 10.1016/j.amc.2015.06.123
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    References listed on IDEAS

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    1. Qi, Feng & Zheng, Miao-Miao, 2015. "Explicit expressions for a family of the Bell polynomials and applications," Applied Mathematics and Computation, Elsevier, vol. 258(C), pages 597-607.
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    Cited by:

    1. Xu, Ai-Min & Cen, Zhong-Di, 2015. "Closed formulas for computing higher-order derivatives of functions involving exponential functions," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 136-141.
    2. Feng Qi & Bai-Ni Guo, 2016. "Some Determinantal Expressions and Recurrence Relations of the Bernoulli Polynomials," Mathematics, MDPI, vol. 4(4), pages 1-11, November.
    3. Zurab K. Silagadze, 2019. "The Basel Problem: A Physicist’s Solution," The Mathematical Intelligencer, Springer, vol. 41(3), pages 14-18, September.

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