A discrete Faa di Bruno's formula

TL;DR

This paper derives formulas describing how finite differences behave under the composition of vector-valued functions, extending classical calculus rules to discrete settings.

Contribution

It introduces a discrete analogue of Faa di Bruno's formula for finite differences of composed vector functions.

Findings

Formulas for finite differences of composed functions

Extension of Faa di Bruno's formula to discrete case

Applications to difference calculus and discrete analysis

Abstract

We derive some formulas that rule the behaviour of finite differences under composition of functions with vector values and arguments.