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