Arithmetic properties of generalized Bernoulli numbers. (English) Zbl 0125.02202

Carlitz, Leonard

J. Reine Angew. Math. 202, 174-182 (1959).

Let \(f\) be a fixed integer \(\geq 1\) and \(\chi(r)\) a primitive character \(\text{mod}\,f\). H.-W. Leopoldt [Abh. Math. Semin. Univ. Hamb. 22, 131–140 (1958; Zbl 0080.03002)] has defined numbers \(B_\chi{}^n,B_\chi{}^n(x)\) by means of \[ \sum_{r=1}^f\chi(r)\frac{te^{(r+x)t}}{e^{ft}-1}=\sum_{n=1}^\infty B_\chi{}^n(x)\frac{t^n}{n!},\quad B_\chi{}^n=B_\chi{}^n(0). \] For \(f=1\), \(\chi\) is the principal character, and \(B_\chi{}^n\) reduces to the ordinary Bernoulli numbers; for \(f=4\) and \(\chi\) the non-principal character \((\text{mod}\,4)\), \(B_\chi{}^n\) reduces to \(-{\textstyle\frac 1{2}}(n+1)E_{n+1}\), where \(E_{n+1}\) is an Euler number. In the present paper the author first extends one of Leopoldt’s results by showing that a theorem of the Staudt-Clausen type holds for the number \(n^{-1}B_\chi{}^n\); he also establishes the related result that if \(p\) is a rational prime such that \(p^e|n\) but \(p\nmid f\), then \(p^e\) divides the numerator of \(B_\chi{}^n\). Next he derives congruences of the Kummer type for \(B_\chi{}^n\). Finally he proves that if \(p^{e-1}(p-1)|m\) but \(p\nmid f\), then \[\frac 1{m+1}B_\chi{}^{m+1}\equiv\frac 1{f}(1-\chi(p))\sum_{r=1}^fr\chi(r)\ (\text{mod}\,p^e). (*)\] In particular for \(f=4\), \((*)\) reduces to a known result for Euler numbers.

Reviewer: A. L. Whiteman (MR 22# 20)

MSC:

11B68

Bernoulli and Euler numbers and polynomials

Keywords:

generalized Bernoulli numbers; Staudt-Clausen type theorem; Euler numbers

Citations:

Zbl 0080.03002