A Supergeometric Fa\`{a} di Bruno Formula

TL;DR

This paper extends the multivariate Faà di Bruno formula to supergeometry, incorporating anticommuting variables and signs, and provides a combinatorial formula for super Bell polynomials.

Contribution

It introduces a supergeometric version of the Faà di Bruno formula, accounting for signs due to odd variables, and derives an explicit combinatorial expression for super Bell polynomials.

Findings

Extended Faà di Bruno formula to supergeometry with signs

Derived explicit combinatorial formula for super Bell polynomials

Demonstrated applicability to generalized super Bell polynomials

Abstract

We extend the multivariate Fa\`{a} di Bruno formula to the super case, where anticommuting odd coordinates are considered. The formula takes the same form as the classical case but contains some nontrivial signs, which essentially measure the failure to order the odd factors and derivatives optimally. As a quick application, we obtain an explicit combinatorial formula for the generalized super Bell polynomials, defined by Fan and Hon.