Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.3

Higher Order Derivatives of Trigonometric Functions, Stirling Numbers of the Second Kind, and Zeon Algebra

Antônio Francisco Neto
DEPRO, Escola de Minas
Campus Morro do Cruzeiro, UFOP
35400-000 Ouro Preto MG


In this work we provide a new short proof of closed formulas for the n-th derivative of the cotangent and secant functions using simple operations in the context of the Zeon algebra. Our main ingredients in the proof comprise a representation of the ordinary derivative as an integration over the Zeon algebra, a representation of the Stirling numbers of the second kind as a Berezin integral, and a change of variables formula under Berezin integration. The approach described here is also suitable to give closed expressions for higher order derivatives of tangent, cosecant and all the aforementioned functions hyperbolic analogues.

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(Concerned with sequence A008277.)

Received April 13 2014; revised versions received July 8 2014; July 16 2014; August 6 2014. Published in Journal of Integer Sequences, August 12 2014.

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