Difference between revisions of "Q-number when a=n is a natural number"
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==Theorem== | ==Theorem== | ||
The following formula holds for $n \in \{1,2,\ldots\}$ and $q \in \mathbb{C} \setminus \{0,1\}$: | The following formula holds for $n \in \{1,2,\ldots\}$ and $q \in \mathbb{C} \setminus \{0,1\}$: | ||
| − | $$[n]_q=\displaystyle\sum_{k=1}^n q^{k-1} | + | $$[n]_q=\displaystyle\sum_{k=1}^n q^{k-1},$$ |
| + | where $[n]_q$ denotes a [[q-number|$q$-number]]. | ||
==Proof== | ==Proof== | ||
Latest revision as of 20:17, 18 December 2016
Theorem
The following formula holds for $n \in \{1,2,\ldots\}$ and $q \in \mathbb{C} \setminus \{0,1\}$: $$[n]_q=\displaystyle\sum_{k=1}^n q^{k-1},$$ where $[n]_q$ denotes a $q$-number.
Proof
References
- 2012: Thomas Ernst: A Comprehensive Treatment of q-Calculus ... (previous) ... (next): ($6.2$)
