On the properties of invariant functions

TL;DR

This paper systematically investigates invariant functions, a class of functions satisfying a specific summation property, and explores their connections to special functions like Bernoulli polynomials, Gamma, and Hurwitz zeta functions.

Contribution

It introduces a systematic study of invariant functions and reveals their relationships with well-known special functions.

Findings

Invariant functions satisfy a key summation property.

Connections between invariant functions and Bernoulli polynomials, Gamma, Hurwitz zeta functions.

Provides a framework for understanding properties of these functions.

Abstract

If $f(x,y)$ is a real function satisfying $y>0$ and $\sum_{r=0}^{n-1}f(x+ry,ny)=f(x,y)$ for $n=1,2,3,\ldots$, we say that $f(x,y)$ is an invariant function. Many special functions including Bernoulli polynomials, Gamma function and Hurwitz zeta function are related to invariant functions. In this paper we systematically investigate the properties of invariant functions.