On the properties of alternating invariant functions

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.