论文标题
关于非阳性整数在Barnes Zeta函数的较高衍生物的值
On values of the higher derivatives of the Barnes zeta function at non-positive integers
论文作者
论文摘要
令$ x $为一个具有正实际零件的复杂数字，$ w_1，\ ldots，w_n $为正理性数字。我们表明，$ w^sζ_n（s，x \ | \ w_1，\ ldots，w_n）$可以表示为hurwitz zeta在$ \ mathbb q（x）$上函数的有限线性组合，其中$ q）由$ w_1，\ ldots，w_n $明确确定。 Furthermore, we give generalizations of Kummer's formula on the gamma function and Koyama-Kurokawa's formulae on the multiple gamma functions, and an explicit formula for the values at non-positive integers for higher order derivatives of the Barnes zeta function in the case that $x$ is a positive rational number, involving the generalized Stieltjes constants and the values at positive integers of the Riemann Zeta功能。我们的公式还可以在$ W_1，\ ldots，w_n $和$ x $的情况下计算近似值。
Let $x$ be a complex number which has a positive real part, and $w_1,\ldots,w_N$ be positive rational numbers. We show that $w^s ζ_N (s, x \ |\ w_1,\ldots, w_N)$ can be expressed as a finite linear combination of the Hurwitz zeta functions over $\mathbb Q(x)$, where $ζ_N (s,x \ |\ w_1,\ldots, w_N)$ is the Barnes zeta function and $w$ is a positive rational number explicitly determined by $w_1,\ldots, w_N$. Furthermore, we give generalizations of Kummer's formula on the gamma function and Koyama-Kurokawa's formulae on the multiple gamma functions, and an explicit formula for the values at non-positive integers for higher order derivatives of the Barnes zeta function in the case that $x$ is a positive rational number, involving the generalized Stieltjes constants and the values at positive integers of the Riemann zeta function. Our formulae also makes it possible to calculate an approximation in the case that $w_1, \ldots, w_N$ and $x$ are positive real numbers.
