Zeon Algebra and Combinatorial Identities

Abstract

Abstract. We show that the ordinary derivative of a real analytic function of one variable can be realized as a Grassmann-Berezin-type integration over the Zeon algebra, the Z-integral. 1. Introduction. The intersection of methods coming from distinct disciplines, such as combinatorics, physics, and graph theory, has been a subject of considerable interest. Clearly, once a common language is established among distinct disciplines, one can gain insight into the structure of each one, first enhancing our understanding through basic manipulations and then, ultimately, pointing toward further constructions. In this manner, some remarkable results have been achieved, including operator methods in graph enumeration problems [1,2,3,4], partitions and quantum field theory [5,6], normal ordering of operators [7,8,9,10], coherent states and combinatorics [11,12], supersymmetry and combinatorics [13], spinorial formulation of graphs [14,15], the Lagrange-Good formula via methods of quantum field theory [16], Heisenberg algebras and Fibonacci numbers [17], and generalizations of the Kirchhoff matrix-tree theorem via Grassmann algebras (see [18] and references therein). Of special relevance is the Zeon algebra [19,20,21,22], comprising important results in the realm of enumerative combinatorics, such as counting problems in lattice partitions [19,20] and derangements [22].In this work, seeking such cross-fertilization of ideas, we revisit a realization of ordinary differentiation of a real analytic function of one variable as a GrassmannBerezin (G-B) integral [23,24,25,26,27,28] over the Zeon algebra introduced in [29]

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