\(q\)-analogue of a binomial coefficient congruence
type: Item
instance of: scholarly article
title: \(q\)-analogue of a binomial coefficient congruence
zbMATH Open document ID: 0816.05009
DOI: 10.1155/S016117129500024X
author: W. Edwin Clark
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1995-06-27T00:00:00Z
full work available at URL: https://eudml.org/doc/47180
review text: If \(p\) is a prime and \(a,b> 0\) are positive integers, the following congruence \[ \left(\begin{smallmatrix} pa\\ pb\end{smallmatrix}\right)= \left(\begin{smallmatrix} a\\ b\end{smallmatrix}\right)(\text{mod }p^ 2) \] is well known. In this note a \(q\)-analogue of the above binomial coefficient is established. The main result is as follows: If \(a\), \(b\), \(n\) are positive integers with \(a> 1\), then \[ \left[\begin{smallmatrix} na\\ nb\end{smallmatrix}\right] (q)= \left[\begin{smallmatrix} a\\ b\end{smallmatrix}\right] (q^{n^ 2}) (\text{mod }\phi(q)^ 2) \] where \(\left[\begin{smallmatrix} n\\ k\end{smallmatrix}\right] (q)\) is the \(q\)-binomial coefficient and \(\phi_ n(q)\) represents the \(n\)th cyclotomic polynomial in the variable \(q\).
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11P83
zbMATH DE Number: 733312
zbMATH Keywords: congruence
zbMATH Keywords: \(q\)-analogue
zbMATH Keywords: binomial coefficient
zbMATH Keywords: cyclotomic polynomial
reviewed by: D. V. Chopra
MaRDI profile type: Publication
OpenAlex ID: W2146410831
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description: scientific article; zbMATH DE number 733312