Q-number

Let $a \in \mathbb{C}$ and $q \in \mathbb{C} \setminus \{0,1\}$. Define the $q$-number $[a]_q$ by $$[a]_q=\dfrac{1-q^a}{1-q}.$$ We define $[0]_q=0$ and if $a_n \in \left\{1,2,\ldots \right\}$, then we get $$[n]_q = \displaystyle\sum_{k=1}^n q^{k-1}.$$

References

• 2012: Thomas Ernst: A Comprehensive Treatment of q-Calculus ... (previous): (6.1)