Difference between revisions of "Q-number"
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Let $a \in \mathbb{C}$ and $q \in \mathbb{C} \setminus \{0,1\}$. Define the $q$-number $[a]_q$ by | Let $a \in \mathbb{C}$ and $q \in \mathbb{C} \setminus \{0,1\}$. Define the $q$-number $[a]_q$ by | ||
$$[a]_q=\dfrac{1-q^a}{1-q}.$$ | $$[a]_q=\dfrac{1-q^a}{1-q}.$$ | ||
=Properties= | =Properties= | ||
| − | [[q-number when a=n is a natural number]]<br /> | + | [[q-number when a=n is a natural number|$q$-number when $a=n$ is a natural number]]<br /> |
| − | [[q- | + | [[q-number of a negative|$q$-number of a negative]]<br /> |
| + | [[1/q-number as a q-number]]<br /> | ||
| + | =See Also= | ||
| + | [[q-factorial|$q$-factorial]]<br /> | ||
| + | |||
| + | =Notes= | ||
| + | [http://mathworld.wolfram.com/q-Bracket.html Mathworld] calls $[a]_q$ the $q$-bracket | ||
=References= | =References= | ||
| − | * {{BookReference|A Comprehensive Treatment of q-Calculus|2012|Thomas Ernst|prev=findme|next=q-number when a=n is a natural number}}: (6.1) | + | * {{PaperReference|q-exponential and q-gamma functions. I. q-exponential functions|1994|D.S. McAnally|prev=Q-derivative power rule|next=findme}} $(2.3)$ (calls $[a]_q$ $(a)_q$) |
| + | * {{BookReference|Special Functions|1999|George E. Andrews|author2=Richard Askey|author3=Ranjan Roy|prev=findme|next=q-factorial}} | ||
| + | * {{BookReference|Quantum Calculus|2002|Victor Kac|author2=Pokman Cheung||prev=findme|next=findme}} $(1.9)$ | ||
| + | * {{BookReference|Quantum Calculus|2002|Victor Kac|author2=Pokman Cheung||prev=findme|next=findme}} $(3.8)$ | ||
| + | * {{BookReference|A Comprehensive Treatment of q-Calculus|2012|Thomas Ernst|prev=findme|next=q-number when a=n is a natural number}}: ($6.1$) (calls $[a]_q$ $\{a\}_q$) | ||
| + | |||
[[Category:SpecialFunction]] | [[Category:SpecialFunction]] | ||
Latest revision as of 04:17, 26 December 2016
Let $a \in \mathbb{C}$ and $q \in \mathbb{C} \setminus \{0,1\}$. Define the $q$-number $[a]_q$ by $$[a]_q=\dfrac{1-q^a}{1-q}.$$
Properties
$q$-number when $a=n$ is a natural number
$q$-number of a negative
1/q-number as a q-number
See Also
Notes
Mathworld calls $[a]_q$ the $q$-bracket
References
- D.S. McAnally: q-exponential and q-gamma functions. I. q-exponential functions (1994)... (previous)... (next) $(2.3)$ (calls $[a]_q$ $(a)_q$)
- 1999: George E. Andrews, Richard Askey and Ranjan Roy: Special Functions ... (previous) ... (next)
- 2002: Victor Kac and Pokman Cheung: Quantum Calculus ... (previous) ... (next) $(1.9)$
- 2002: Victor Kac and Pokman Cheung: Quantum Calculus ... (previous) ... (next) $(3.8)$
- 2012: Thomas Ernst: A Comprehensive Treatment of q-Calculus ... (previous) ... (next): ($6.1$) (calls $[a]_q$ $\{a\}_q$)
