Difference between revisions of "Q-number"

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Let $q \in \mathbb{C} \setminus \{1\}$ and define the $q$ numbers
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$$[0]_0=0$$
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Let $a \in \mathbb{C}$ and $q \in \mathbb{C} \setminus \{0,1\}$. Define the $q$-number $[a]_q$ by
and for $n>0$ a positive integer,
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$$[a]_q=\dfrac{1-q^a}{1-q}.$$
$$[n]_q=\dfrac{1-q^n}{1-q}=1+q+q^2+\ldots+q^{n-1}.$$
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=Properties=
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[[q-number when a=n is a natural number|$q$-number when $a=n$ is a natural number]]<br />
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[[q-number of a negative|$q$-number of a negative]]<br />
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[[1/q-number as a q-number]]<br />
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=See Also=
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[[q-factorial|$q$-factorial]]<br />
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=Notes=
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[http://mathworld.wolfram.com/q-Bracket.html Mathworld] calls $[a]_q$ the $q$-bracket
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=References=
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* {{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 $[a]_q$ $(a)_q$)
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* {{BookReference|Special Functions|1999|George E. Andrews|author2=Richard Askey|author3=Ranjan Roy|prev=findme|next=q-factorial}}
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* {{BookReference|Quantum Calculus|2002|Victor Kac|author2=Pokman Cheung||prev=findme|next=findme}} $(1.9)$
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* {{BookReference|Quantum Calculus|2002|Victor Kac|author2=Pokman Cheung||prev=findme|next=findme}} $(3.8)$
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* {{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 $[a]_q$ $\{a\}_q$)
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[[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

$q$-factorial

Notes

Mathworld calls $[a]_q$ the $q$-bracket

References