Divided Differences of Implicit Functions

TL;DR

This paper develops a mathematical framework to express divided differences of implicitly defined functions in terms of the divided differences of the defining function, generalizing previous results on inverse functions.

Contribution

It introduces a general method to relate divided differences of implicit functions to those of their defining functions, extending existing theories.

Findings

Provides explicit formulas for divided differences of implicit functions.

Generalizes previous results on inverse functions.

Enhances computational techniques for implicit function analysis.

Abstract

Under general conditions, the equation $g(x,y) = 0$ implicitly defines $y$ locally as a function of $x$. In this article, we express divided differences of $y$ in terms of bivariate divided differences of $g$, generalizing a recent result on divided differences of inverse functions.