Geodesic
Higher order derivatives of trigonometric functions, Stirling numbers of the second kind, and Zeon algebra
Journal of integer sequences, Tome 17 (2014) no. 9
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.
Classification : 11B73, 33B10, 05A15, 05A18, 05A19
Keywords: zeon algebra, Berezin integration, cotangent, secant, Stirling number of the second kind, generating function 
@article{JIS_2014__17_9_a0,
     author = {Neto,  Ant\^onio Francisco},
     title = {Higher order derivatives of trigonometric functions, {Stirling} numbers of the second kind, and {Zeon} algebra},
     journal = {Journal of integer sequences},
     year = {2014},
     volume = {17},
     number = {9},
     zbl = {1358.11044},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2014__17_9_a0/}
}
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JO  - Journal of integer sequences
PY  - 2014
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IS  - 9
UR  - http://geodesic.mathdoc.fr/item/JIS_2014__17_9_a0/
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%0 Journal Article
%A Neto,  Antônio Francisco
%T Higher order derivatives of trigonometric functions, Stirling numbers of the second kind, and Zeon algebra
%J Journal of integer sequences
%D 2014
%V 17
%N 9
%U http://geodesic.mathdoc.fr/item/JIS_2014__17_9_a0/
%G en
%F JIS_2014__17_9_a0
Neto,  Antônio Francisco. Higher order derivatives of trigonometric functions, Stirling numbers of the second kind, and Zeon algebra. Journal of integer sequences, Tome 17 (2014) no. 9. http://geodesic.mathdoc.fr/item/JIS_2014__17_9_a0/