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 $n=1,2,3,\dots$, 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.

中文翻译：

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关于不变函数的性质

如果$F（X,y）$是一个实数函数，满足$y>0$和${\Sigma }_{r=0}^{n-1}F（X+ry,ny）=F（X,y）$为了$n=1,2,3,\dots \dots$，我们说$F（X,y）$是一个不变函数。许多特殊函数包括伯努利多项式、Gamma函数和Hurwitz zeta函数都与​​不变函数有关。在本文中，我们系统地研究了不变函数的性质。