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On the properties of invariant functions (2209.14625v1)
Published 29 Sep 2022 in math.CA
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.