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q-Binomial Coefficient

A q-Analog for the Binomial Coefficient, also called the Gaussian Coefficient. It is given by

{n\choose m}_q\equiv {(q)_n\over (q)_m (q)_{n-m}}=\prod_{i=0}^{j-1}{1-q^{k-i}\over 1-q^{i+1}},
\end{displaymath} (1)

(q)_k\equiv \prod_{m=1}^\infty {1-q^m\over 1-q^{k+m}}.
\end{displaymath} (2)

For example, the first few $q$-binomial coefficients are
$\displaystyle {2\choose 1}_q$ $\textstyle =$ $\displaystyle {1-q^2\over 1-q}=1+q$ (3)
$\displaystyle {3\choose 1}_q$ $\textstyle =$ $\displaystyle {3\choose 2}_q={1-q^3\over 1-q}=1+q+q^2$ (4)
$\displaystyle {4\choose 1}_q$ $\textstyle =$ $\displaystyle {4\choose 3}_q={1-q^4\over 1-q}=1+q+q^2+q^3$ (5)
$\displaystyle {4\choose 2}_q$ $\textstyle =$ $\displaystyle {(1-q^3)(1-q^4)\over(1-q)(1-q^2)}=(1+q)(1+q+q^2).$ (6)

From the definition, it follows that

{n\choose 1}_q={n\choose n-1}_q=\sum_{i=0}^{n-1} q^n.
\end{displaymath} (7)

In the Limit $q\to 1$, the $q$-binomial coefficient collapses to the usual Binomial Coefficient.

See also Cauchy Binomial Theorem, Gaussian Polynomial

© 1996-9 Eric W. Weisstein