A generating function of higher-dimensional Apostol-Zagier sums and its reciprocity law
A generating function of higher-dimensional Apostol-Zagier sums and its reciprocity law
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DOI：
10.1016/j.jnt.2005.05.008
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发表时间：
2006-03
影响因子：
0.7
通讯作者：
S. Fukuhara;N. Yui
中科院分区：
文献类型：
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作者：
S. Fukuhara;N. Yui
引入具有复参数z的高维Dedekind和，推广了Zagier的高维Dedekind和。当z→∞时，和趋向于Zagier的高维Dedekind和。我们证明了这些和是高维apostoll - Zagier和的生成函数，这些apostoll - Zagier和被定义为Apostol和与Zagier和的杂交。我们证明了和的互易律。新的互易律包含了Apostol和Zagier和的互易公式作为特例。此外，作为它的应用，我们得到了Hurwitz zeta函数的特殊值与伯努利数之间的关系，以及新的三角恒等式。
We introduce higher-dimensional Dedekind sums with a complex parameter z, generalizing Zagier's higher-dimensional Dedekind sums. The sums tend to Zagier's higher-dimensional Dedekind sums as z→∞. We show that the sums turn out to be generating functions of higher-dimensional Apostol–Zagier sums which are defined to be hybrids of Apostol's sums and Zagier's sums. We prove reciprocity law for the sums. The new reciprocity law includes reciprocity formulas for both Apostol and Zagier's sums as its special case. Furthermore, as its application we obtain relations between special values of Hurwitz zeta function and Bernoulli numbers, as well as new trigonometric identities.