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Combinatorial congruences modulo prime powers

Authors: Zhi-Wei Sun and Donald M. Davis
Journal: Trans. Amer. Math. Soc. 359 (2007), 5525-5553
MSC (2000): Primary 11B65; Secondary 05A10, 11A07, 11B68, 11S05
Published electronically: May 1, 2007
MathSciNet review: 2327041
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ p$ be any prime, and let $ \alpha $ and $ n$ be nonnegative integers. Let $ r\in \mathbb{Z}$ and $ f(x)\in \mathbb{Z}[x]$. We establish the congruence

$\displaystyle p^{\deg f}\sum _{k\equiv r\, (\operatorname{mod}p^{\alpha })} \bi... ...atorname{mod} p^{\sum _{i=\alpha }^{\infty } \lfloor n/{p^{i}}\rfloor }\right )$

(motivated by a conjecture arising from algebraic topology) and obtain the following vast generalization of Lucas' theorem: If $ \alpha $ is greater than one, and $ l,s,t$ are nonnegative integers with $ s,t<p$, then

\begin{displaymath}\begin{split} &\frac{1}{\lfloor n/p^{\alpha -1}\rfloor !} \su... ...p^{\alpha -1}}\right )^{l} (\operatorname{mod} p). \end{split}\end{displaymath}

We also present an application of the first congruence to Bernoulli polynomials and apply the second congruence to show that a $ p$-adic order bound given by the authors in a previous paper can be attained when $ p=2$.

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Additional Information

Zhi-Wei Sun
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China

Donald M. Davis
Affiliation: Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015

Received by editor(s): September 6, 2005
Received by editor(s) in revised form: November 26, 2005
Published electronically: May 1, 2007
Additional Notes: The first author is responsible for communications, and partially supported by the National Science Fund for Distinguished Young Scholars (Grant No. 10425103) in People’s Republic of China.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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