Properties of q-Binomial Coefficients

Abstract

Let us examine some properties of the q-binomial coefficients, defined by (4.5), with n and j being nonnegative integers and n ≥ j. Because we will recover the ordinary binomial coefficients if we take q → 1, we expect their q-analogues to have similar properties. Firstly, as already remarked in (5.4),

$$ \left[ {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right] = \frac{{[n]!}} {{[j]![n - j]!}} = \left[ {\begin{array}{*{20}c} n \\ {n - j} \\ \end{array} } \right] $$

((6.1))

follows exactly the classical result. However, the correspondence is more subtle for another identity of binomial coefficients, the Pascal rule:

$$ \left( {\begin{array}{*{20}c} n \\ j \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {n - 1} \\ {j - 1} \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {n - 1} \\ j \\ \end{array} } \right), 1 \leqslant j \leqslant n - 1. $$