q-Analogue of (x − a)ⁿ, n an Integer, and q-Derivatives of Binomials

Abstract

As remarked in Chapter 1, D _q is a linear Operator on the space of polynomials. We shall try to apply Theorem 2.1 to D ≡ D _q . We shall need for that the following q-analogue of n!:

$$ [n]! = \left\{ {\begin{array}{*{20}c} {1 if n = 0,} \\ {[n] \times [n - 1] \times \cdots \times [1] if n = if n = 1,2, \ldots .} \\ \end{array} } \right. $$

((3.1))