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An extension of Lucas' theorem


Authors: Hong Hu and Zhi-Wei Sun
Journal: Proc. Amer. Math. Soc. 129 (2001), 3471-3478
MSC (2000): Primary 11B39; Secondary 11A07, 11B65
DOI: https://doi.org/10.1090/S0002-9939-01-06234-7
Published electronically: June 8, 2001
MathSciNet review: 1860478
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Abstract | References | Similar Articles | Additional Information

Abstract:

Let $p$ be a prime. A famous theorem of Lucas states that $\binom {mp+s}{np+t}\equiv \binom mn\binom st (\operatorname{mod} p)$ if $m,n,s,t$ are nonnegative integers with $s,t<p$. In this paper we aim to prove a similar result for generalized binomial coefficients defined in terms of second order recurrent sequences with initial values $0$ and $1$.


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Additional Information

Hong Hu
Affiliation: Department of Mathematics, Huaiyin Normal College, Huaiyin 223001, Jiangsu Province, People’s Republic of China

Zhi-Wei Sun
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, People’s Republic of China
Email: zwsun@nju.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-01-06234-7
Received by editor(s): April 18, 2000
Published electronically: June 8, 2001
Additional Notes: The second author is responsible for all the communications, and supported by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, and the National Natural Science Foundation of P. R. China.
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2001 American Mathematical Society

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