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

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

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 $[a]_q$ $\{a\}_q$)

@@ Line 1: / Line 1: @@
-Let $q \in \mathbb{C} \setminus \{1\}$ and define the $q$ numbers
+__NOTOC__
-$$[0]_0=0$$
+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,
+$$[a]_q=\dfrac{1-q^a}{1-q}.$$
-$$[n]_q=\dfrac{1-q^n}{1-q}=1+q+q^2+\ldots+q^{n-1}.$$
+=Properties=
+[[q-number when a=n is a natural number|$q$-number when $a=n$ is a natural number]]<br />
+[[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=
+* {{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]]