Arithmetic Properties of Generalized Bernoulli Numbers.
Arithmetic Properties of Generalized Bernoulli Numbers.
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广义伯努利数的算术性质。
DOI：
10.1515/crll.1959.202.174
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发表时间：
1959
期刊：
影响因子：
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通讯作者：
L. Carlitz
中科院分区：
文献类型：
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作者：
L. Carlitz
(1. 3) Bl(x] = Σ(\ Byx~ = (Βχ + x)».r = 0 \ ' / 对于 / == l，χ 为主要特征，B" 化简为普通伯努利数；对于 / = 4 且 χ 为非主要特征 (mod 4)，BTM 化简为 — — ̂ — ^n + i ? 其中 En+1 为欧拉数。主要结果Leopoldt 的论文是数字 BTM 的 Staudt-Clausen 定理的类比。本文首先涉及 Leopoldt 的结果的扩展。我们证明 Staudt-Clausen 类型的定理对于数字 l — B^\ 成立，我们还得到了相关结果，即如果 p 是有理素数，使得 p\ n 但 p \ f IV，则 p 可以除以 Br 的分子。我们证明（定理 5）BTM 满足 Kummer 类型的同余式 最后我们证明如果 p~(p — l)\m 但 p t /，则
(1. 3) Bl(x] = Σ(\ Byx~ = (Βχ + x)». r = 0 \ ' / For / == l, χ is the principal character, and B" reduces to the ordinary Bernoulli numbers; for / = 4 and χ the non-principal character (mod 4), BTM reduces to — — ̂ — ^n + i ? where En+1 is an Euler number. The main result of Leopoldt's paper is an analog of the Staudt-Clausen theorem for the numbers BTM. The present paper is concerned, in the first place, with an extension of Leopoldt's rriain result. We show that a theorern of the Staudt-Clausen type holds for the number l — B^\ we have also the related result that if p is a rational prime such that p\ n but p \ f IV then p divides the numerator of Br In the next place we show (Theorem 5) that BTM satisfies congruences of the Kummer type. Finally we show that if p~(p — l)\m but p t /, then