Abstract We prove that the Weyl fractional derivative is a useful instrument to express certain properties of the zeta related functions. Specifically, we show that a known reflection property of the Hurwitz zeta function ζ(n, a) of integer first argument can be extended to the more general case of ζ(s, a), with complex s, by replacement of the ordinary derivative of integer order by Weyl fractional derivative of complex order. Besides, ζ(s, a) with ℜ(s) > 2 is essentially the Weyl (s − 2)-derivative of ζ(2, a). These properties of the Hurwitz zeta function can be immediately transferred to a family of polygamma functions of complex order defined in a natural way. Finally, we discuss the generalization of a recently unveiled reflection property of the Lerch’s transcendent.

中文翻译：

---

zeta 相关函数在分数阶导数方面的反射特性

摘要 我们证明了 Weyl 分数阶导数是表达 zeta 相关函数的某些性质的有用工具。具体来说，我们表明整数第一个参数的 Hurwitz zeta 函数 ζ(n, a) 的已知反射性质可以扩展到 ζ(s, a) 的更一般情况，具有复数 s，通过替换普通导数整数阶由复阶的 Weyl 分数阶导数。此外，ζ(s, a) ℜ(s) > 2 本质上是 ζ(2, a) 的 Weyl (s − 2) 导数。Hurwitz zeta 函数的这些性质可以立即转移到以自然方式定义的复阶多伽马函数族中。最后，我们讨论了最近公布的 Lerch 超验的反射属性的概括。