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SOME PROPERTIES OF A SEQUENCE ANALOGOUS TO EULER NUMBERS

Part of: Elementary number theory

Published online by Cambridge University Press:  12 June 2012

ZHI-HONG SUN
ZHI-HONG SUN*
Affiliation:
School of Mathematical Sciences, Huaiyin Normal University, Huaian, Jiangsu 223001, PR China (email: zhihongsun@yahoo.com)
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Abstract

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Let $\{U_n\}$ be given by $U_0=1$ and $U_n=-2\sum _{k=1}^{[n/2]} \binom n{2k}U_{n-2k}\ (n\ge 1)$, where $[\cdot ]$ is the greatest integer function. Then $\{U_n\}$ is analogous to the Euler numbers and $U_{2n}=3^{2n}E_{2n}(\frac 13)$, where $E_m(x)$ is the Euler polynomial. In a previous paper we gave many properties of $\{U_n\}$. In this paper we present a summation formula and several congruences involving $\{U_n\}$.

Keywords

congruencesummation formulaEuler number

MSC classification

Secondary: 11A07: Congruences; primitive roots; residue systems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 3 , June 2013 , pp. 425 - 440
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc. 

References

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