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On the Differential Equation of First and Second Order in the Zeon Algebra

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Abstract

In this paper, the study of linear differential equations involving one conventional and two nilpotent variables is started. This is a natural extension of the case of one involved nilpotent (para-Grassmann) variable studied earlier. In the case considered here, the two nilpotent variables are assumed to commute, hence they are generators of a (generalized) zeon algebra. Using the natural para-supercovariant derivatives \(D_i\) transferred from the study of a para-Grassmann variable, we consider linear differential equations of order at most two in \(D_i\) and discuss the structure of their solutions. For this, convenient graphical representations in terms of simple graphs are introduced.

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Notes

  1. We use here the notation [[k]] instead of the usual [k] to avoid confusion with the q-deformed numbers \([k]_q\).

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  1. Department of Mathematics, University of Haifa, Haifa, 3498838, Israel

    Toufik Mansour

  2. Im Haindell 99, 65843, Sulzbach, Germany

    Matthias Schork

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  1. Toufik Mansour
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Mansour, T., Schork, M. On the Differential Equation of First and Second Order in the Zeon Algebra. Adv. Appl. Clifford Algebras 31, 21 (2021). https://doi.org/10.1007/s00006-021-01126-7

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