Q-number

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

• 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 $[a]_q$ $\{a\}_q$)