Taylor Expansion of homogeneous functions

TL;DR

This paper derives a specialized Taylor polynomial for positive homogeneous functions of order m, showing it simplifies to a polynomial in the variable itself rather than in the difference, with applications to functions of order 1.

Contribution

It introduces a new form of Taylor polynomial for positive homogeneous functions, extending classical Taylor expansion to this class of functions.

Findings

The Taylor polynomial for positive homogeneous functions is a polynomial in the variable, not in the difference.

The polynomial's order matches the function's homogeneity order.

Applicable to powers of homogeneous functions of order 1.

Abstract

We derive the Taylor polynomial of a function, which is $m$-times continuously differentiable and positive homogeneous of order $m$. The Taylor polynomial in $a$ for $f(b)$ of order $m$ in general is a polynomial of order $m$ in $b-a$. If the given function is positive homogeneous of order $m$, the Taylor polynomial is a polynomial in $b$ rather than $b-a$, and the order of all terms is $m$. The result can be applied to powers of homogeneous functions of order $1$ as well.