A generalization of the Gaussian formula and a q-analog of Fleck's congruence

JOURNAL OF NUMBER THEORY

卷:133

Article

Schultz, Andrew¹  Walker, Robert² 

[1] Wellesley Coll, Dept Math, Wellesley, MA 02481 USA

[2] Univ Michigan, Dept Math, Ann Arbor, MI 48109 USA

关键词: q-Binomial coefficient;    Fleck's congruence;    q-Analog;    Gaussian formula;    Chu-Vandermonde;   

DOI  :  10.1016/j.jnt.2013.05.003

来源: Elsevier

【 摘 要 】

The q-binomial coefficients are the polynomial cousins of the traditional binomial coefficients, and a number of identities for binomial coefficients can be translated into this polynomial setting. For instance, the familiar vanishing of the alternating sum across row n is an element of Z(+) of Pascal's triangle is captured by the so-called Gaussian formula, which states that Sigma(n)(m=0)(-1)(m) ((n)(m))(q) is 0 if n is odd, and is equal to m0 m Pi(k odd)(1 - q(k)) if n is even. In this paper, we find a q-binomial congruence which synthesizes this result and Fleck's congruence for binomial coefficients, which asserts that for n,p is an element of Z(+), with p a prime, Sigma(m j (mod p))(-1)(m) ((n)(m))( 0) (mod p(left perpendicular n-1/p-1 right perpendicular)). (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

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