Some divisibility properties of binomial and q-binomial coefficients

JOURNAL OF NUMBER THEORY

卷:135

Article

Guo, Victor J. W.¹  Krattenthaler, C.² 

[1] E China Normal Univ, Dept Math, Shanghai 200062, Peoples R China

[2] Univ Vienna, Fak Math, A-1090 Vienna, Austria

关键词: Binomial coefficients;    Lucas' theorem;    Euler's totient theorem;    q-binomial coefficients;    Gaussian polynomials;    Catalan numbers;    q-Catalan numbers;    Positive polynomials;   

DOI  :  10.1016/j.jnt.2013.08.012

来源: Elsevier

【 摘 要 】

We first prove that if a has a prime factor not dividing b then there are infinitely many positive integers n such that ((an+bn)(an)) is not divisible by bn + 1. This confirms a recent conjecture of Z.-W. Sun. Moreover, we provide some new divisibility properties of binomial coefficients: for example, we prove that ((12n)(3n)) and ((12n)(4n)) are divisible by 6n - 1, and that ((330n)(88n)) is divisible by 66n - 1, for all positive integers n. As we show, the latter results are in fact consequences of divisibility and positivity results for quotients of q-binomial coefficients by q-integers, generalising the positivity of q-Catalan numbers. We also put forward several related conjectures. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free