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Gaussian Polynomial

Defined by

\begin{displaymath}[l]\equiv {1-q^l\over 1-q}
\end{displaymath} (1)

for integral $l$, and
\left[{\matrix{n\cr k\cr}}\right]\equiv\cases{
\prod_{l=1}^k {[n-l+1]\over [l]} & for $0\leq k\leq n$\cr
0 & otherwise.\cr}
\end{displaymath} (2)

Unfortunately, the Notation conflicts with that of Gaussian Brackets and the Nearest Integer Function. Gaussian Polynomials satisfy the identities
{\left[{\matrix{n+1\cr k+1\cr}}\right]\over\left[{\matrix{n\cr k+1\cr}}\right]}={1-q^{n+1}\over 1-q^{n-k}}
\end{displaymath} (3)

{\left[{\matrix{n+1\cr k+1\cr}}\right]\over\left[{\matrix{n+1\cr k\cr}}\right]}={1-q^{n-k+1}\over 1-q^{k+1}}.
\end{displaymath} (4)

For $q=1$, the Gaussian polynomial turns into the Binomial Coefficient.

See also Binomial Coefficient, Gaussian Coefficient, q-Series

© 1996-9 Eric W. Weisstein