交变不变函数的性质

IF 0.9 3区数学 Q2 MATHEMATICS, APPLIED

Bulletin des Sciences Mathematiques Pub Date : 2025-09-08 DOI:10.1016/j.bulsci.2025.103724

Haiqing Zhu , Su Hu , Min-Soo Kim

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引用次数: 0

摘要

对于任意正奇整数，满足函数方程∑r=0n−1(−1)rf(x+ry,ny)=f（x,y）的函数称为交替不变函数。这些函数的例子包括欧拉多项式，交替Hurwitz zeta函数和它们相关的Gamma函数。本文系统地研究了交变不变函数的基本性质。证明了这类函数的集合在平移、反射和微分下是封闭的。此外，我们定义了交替不变函数的卷积运算，并分别导出了欧拉多项式和交替Hurwitz zeta函数的显式卷积公式。此外，利用分布关系，我们构造了交替不变函数的新例子，包括三角函数、指数函数和对数函数的适当组合等。

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On the properties of alternating invariant functions

Functions satisfying the functional equation

\sum_{r = 0}^{n - 1} {(- 1)}^{r} f (x + r y, n y) = f (x, y), for any positive odd integer n,

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.