Q-number when a=n is a natural number

Theorem

The following formula holds for $n \in \{1,2,\ldots\}$ and $q \in \mathbb{C} \setminus \{0,1\}$: $$[n]_q=\displaystyle\sum_{k=1}^n q^{k-1},$$ where $[n]_q$ denotes a $q$-number.

References

• 2012: Thomas Ernst: A Comprehensive Treatment of q-Calculus ... (previous) ... (next): ($6.2$)