A Note on Bernoulli Numbers and Shintani Generalized Bernoulli Polynomials
A Note on Bernoulli Numbers and Shintani Generalized Bernoulli Polynomials
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DOI：
10.1090/s0002-9947-96-01479-1
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发表时间：
1996
影响因子：
1.3
通讯作者：
M. Eie
中科院分区：
文献类型：
--
作者：
M. Eie
为了表示全真实的数的Dedekind zeta函数在非正整数处的特殊值，Shintani于1976年引入了广义Bernoulli多项式.这种多项式的系数是伯努利数的乘积的有限组合，这是很难掌握的。另一方面，Zagier能够得到明确的公式的特殊价值的情况下，真实的二次数域。本文将改进Shintani公式，证明特殊值可由有限多项式集确定。这提供了一种方便的方法来评估各种类型的Dedekind函数的特殊值。事实上，作者[4]考虑的一类更广泛的zeta函数对于其特殊值也有类似的公式。因此，我们能够通过zeta函数之间的恒等式找到伯努利数之间的无穷多个恒等式。所有这些身份都很难证明。1. Bernoulli数Bn（n = 0，1，2，…）定义为t _00 Bntn et 1 E n！因为通过直接验证，函数t t et _1 2是t的偶函数。用伯努利数表示黎曼zeta函数的特殊值
Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields. In this paper, we shall improve Shintani's formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [4] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise. 1. IDENTITIES AMONG BERNOULLI NUMBERS The Bernoulli numbers Bn (n = 0,1, 2, ..) are defined by t _00 Bntn et1 E n! ' Itl 1 since the function t t et _1 2 is an even function of t by direct verification. Bernoulli numbers are used to express the special values of Riemann zeta function