Gaussian Polynomial

Defined by

$$[l]\equiv {1-q^l\over 1-q}$$ (1)

for integral $l$, and

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

Unfortunately, the Notation conflicts with that of Gaussian Brackets and the Nearest Integer Function. Gaussian Polynomials satisfy the identities

$${\left[{\matrix{n+1\cr k+1\cr}}\right]\over\left[{\matrix{n\cr k+1\cr}}\right]}={1-q^{n+1}\over 1-q^{n-k}}$$ (3)

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

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

See also Binomial Coefficient, Gaussian Coefficient, q-Series