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Congruences for sequences analogous to Euler numbers | Zhi-Hong Sun
; Hai-Yan Wang
; | Date: |
28 Jul 2013 | Abstract: | For a given real number $a$ we define the sequence ${E_{n,a}}$ by
$E_{0,a}=1$ and $E_{n,a}=-asum_{k=1}^{[n/2]}
inom n{2k}E_{n-2k,a}$ $(nge 1)$, where $[x]$ is the greatest integer not
exceeding $x$. Since $E_{n,1}=E_n$ is the n-th Euler number, $E_{n,a}$ can be
viewed as a natural generalization of Euler numbers. In this paper we deduce
some identities and an inversion formula involving ${E_{n,a}}$, and establish
congruences for $E_{2n,a}mod{2^{{
m ord}_2n+8}}$, $E_{2n,a}pmod{3^{{
m
ord}_3n+5}}$ and $E_{2n,a}pmod{5^{{
m ord}_5n+4}}$ provided that $a$ is a
nonzero integer, where ${
m ord}_pn$ is the least nonnegative integer $alpha$
such that $p^{a}mid n$ but $p^{a+1}
mid n$. | Source: | arXiv, 1307.7370 | Services: | Forum | Review | PDF | Favorites |
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