论文标题
Bernoulli功能的简介
An introduction to the Bernoulli function
论文作者
论文摘要
我们根据J. Jensen建议的积分表示，探索了Zeta函数的变体。 The Bernoulli function $\operatorname{B}(s, v) = - s\, ζ(1-s, v)$ can be introduced independently of the zeta function if it is based on a formula first given by Jensen in 1895. We examine the functional equation of $\operatorname{B}(s, v)$ and its representation by the Riemann $ζ$ and $ξ$ function, and Hadamard，Worpitzky和Hasse的经典结果以$ \ operatatorName {b}（s，v）而言。$扩展的bernoulli函数定义了由Euler在1735年研究的理性数字基于其理性数字的bernoulli数字，这是Euler和Euler和Andre的数字。引入Euler函数作为Hurwitz-Bernoulli函数值之间的差异。 André函数和SEKI函数是扩展Euler resp的未签名版本。 Bernoulli功能。
We explore a variant of the zeta function interpolating the Bernoulli numbers based on an integral representation suggested by J. Jensen. The Bernoulli function $\operatorname{B}(s, v) = - s\, ζ(1-s, v)$ can be introduced independently of the zeta function if it is based on a formula first given by Jensen in 1895. We examine the functional equation of $\operatorname{B}(s, v)$ and its representation by the Riemann $ζ$ and $ξ$ function, and recast classical results of Hadamard, Worpitzky, and Hasse in terms of $\operatorname{B}(s, v).$ The extended Bernoulli function defines the Bernoulli numbers for odd indices basing them on rational numbers studied by Euler in 1735 that underlie the Euler and André numbers. The Euler function is introduced as the difference between values of the Hurwitz-Bernoulli function. The André function and the Seki function are the unsigned versions of the extended Euler resp. Bernoulli function.
