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Extensions of Stern's congruence for Euler numbers | Zhi-Hong Sun
; Long Li
; | Date: |
15 Jul 2013 | Abstract: | For a nonzero integer $a$ let ${E_n^{(a)}}$ be given by
$sum_{k=0}^{[n/2]}inom n{2k}a^{2k}E_{n-2k}^{(a)}=(1-a)^n$ $(n=0,1,2,...)$,
where $[x]$ is the greatest integer not exceeding $x$. As $E_n^{(1)}=E_n$ is
the Euler number, $E_n^{(a)}$ can be viewed as a generalization of Euler
numbers. Let $k$ and $m$ be positive integers, and let $b$ be a nonnegative
integer. In this paper, we determine $E_{2^mk+b}^{(a)}$ modulo $ 2^{m+10}$ for
$mge 5$. For $mge 5$ we also establish congruences for
$U_{kvarphi{(5^m)}+b},; E_{kvarphi{(5^m)}+b},;
S_{kvarphi{(5^m)}+b}pmod{5^{m+5}}$ and $S_{kvarphi{(3^m)}+b}pmod{3^{m+5}},$
where $U_{2n}=E_{2n}^{(3/2)}$, $S_n=E_n^{(2)}$ and $varphi(n)$ is Euler’s
function. | Source: | arXiv, 1307.3902 | Services: | Forum | Review | PDF | Favorites |
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