On the properties of alternating invariant functions

TL;DR

This paper systematically studies alternating invariant functions, revealing their closure properties, defining a convolution operation, and constructing new examples through distributional relations, thus deepening understanding of their structure and applications.

Contribution

It introduces the concept of convolution for alternating invariant functions and constructs new examples, expanding the theoretical framework and applications.

Findings

Set of alternating invariant functions is closed under translation, reflection, and differentiation.

Derived explicit convolution formulas for Euler polynomials and alternating Hurwitz zeta functions.

Constructed new examples using distributional relations, including trigonometric, exponential, and logarithmic functions.

Abstract

Functions satisfying the functional equation \begin{align*} \sum_{r=0}^{n-1} (-1)^r f(x+ry, ny) = f(x,y), \quad \text{for any positive odd integer $n$}, \end{align*} are named the alternating invariant functions. Examples of such functions include Euler polynomials, alternating Hurwitz zeta functions and their associated Gamma functions. In this paper, we systematically investigate the fundamental properties of alternating invariant functions. We prove that the set of such functions is closed under translation, reflection, and differentiation. In addition, we define a convolution operation on alternating invariant functions and derive explicit convolution formulas for Euler polynomials and alternating Hurwitz zeta functions, respectively. Furthermore, using distributional relations, we construct new examples of alternating invariant functions, including suitable combinations of trigonometric, exponential, and logarithmic functions, among others.