Some congruences and identities for Gauss polynomials: A noncommutative technique (Q1184769)

The Gauss polynomial is defined, when \(0<n<m\), by \[ {m\brack n}_ q=\begin{cases} {(q^ m-1)(q^{m-1}-1)\cdots(q^{m-n+1}-1)\over (q-1)(q^ 2- 1)\cdots(q^ n-1)}, \quad & \text{ if }q\neq 1, \\ {m\choose n}, \quad & \text{if \(q=1\) (for the sake of continuity)}. \end{cases} \] The author uses Gauss polynomials to obtain the following generalization of a congruence of Lucas: If \(m=ka+b\) and \(n=kc+d\), where \(0\leq b<k\), \(0\leq d<k\), then \[ {m\brack n}_ q\equiv{a\choose c}{b\brack d}_ d(\bmod \Phi_ k(q)\mathbb{Z}[q]), \] where \(\Phi_ k\) is the \(k\)-th cyclotomic polynomial. Two proofs are given, one of which contains a noncommutative \(q\)-analoge of Newton's formula. Some applications in the form of identities and congruences are given. For example \[ (\Phi_ n(q))^ 2\text{ divides }{2n\brack n}_ q-1-q^{(n^ 2)}. \]