منابع مشابه
A Pretty Binomial Identity
Elementary proofs abound: the first identity results from choosing x = y = 1 in the binomial expansion of (x+y). The second one may be obtained by comparing the coefficient of x in the identity (1 + x)(1 + x) = (1 + x). The reader is surely aware of many other proofs, including some combinatorial in nature. At the end of the previous century, the evaluation of these sums was trivialized by the ...
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Remark. This identity is easily verified using the WZ method, in a generalized form [Z] that applies when the summand is a hypergeometric term times a WZ potential function. It holds for all positive n, since it holds for n=1,2,3 (check!), and since the sequence defined by the sum satisfies a certain (homog.) third order linear recurrence equation. To find the recurrence, and its proof, downloa...
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We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for the same quantity. The goal of this note is to give a simple (and interesting) probabilistic proof of the binomial identity n ∑ k=0 ( n k ) (−1) θ θ + k = n ∏ k=1 k θ + k , for all θ > 0 and all n ∈ ...
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A certain alternating sum u(n) of n + 1 products of two binomial coefficients has a property similar to Wolstenholme’s theorem, namely u(p) ≡ −1 (mod p3) for all primes p ≥ 5. However, this congruence also holds for certain composite integers p which appear to always have exactly two prime divisors, one of which is always 2 or 5. This phenomenon will be partly explained and the composites in qu...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1999
ISSN: 0012-365X
DOI: 10.1016/s0012-365x(98)00178-2