\(q\)-analogue of a binomial coefficient congruence
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 \(q\).
scientific article; zbMATH DE number 733312
scientific article