Zeon algebra and combinatorial identities.

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. As a by-product of this representation, we give new proofs of the Fa`a di Bruno formula and Spivey’s identity [M. Z. Spivey, J. Integer Seq., 11 (2008), 08.2.5], and we recover the representation of the Stirling numbers of the second kind and the Bell numbers of Staples and Schott [European J. Combin., 29 (2008), pp. 1133–1138]. The approach described here is suitable to accommodate new Z-integral representations including Stirling numbers of the first kind, central Delannoy, Euler, Fibonacci, and Genocchi numbers, and the special polynomials of Bell, generalized Bell, Hermite, and Laguerre.