Q-factorial

The $q$-factorial is defined for a non-negative integer $k$ by $$[n]_q! = \displaystyle\prod_{k=1}^n [k]_q= \left( \dfrac{1-q}{1-q} \right) \left( \dfrac{1-q^2}{1-q} \right) \ldots \left( \dfrac{1-q^n}{1-q} \right),$$ where $[k]_q$ denotes a $q$-number.

Properties

Q-derivative power rule

See Also

$q$-number
$q$-Pochhammer

References

• D.S. McAnally: q-exponential and q-gamma functions. I. q-exponential functions (1994)... (previous)... (next)
• 1999: George E. Andrews, Richard Askey and Ranjan Roy: Special Functions ... (previous) ... (next)
• 2002: Victor Kac and Pokman Cheung: Quantum Calculus ... (previous) ... (next) $(3.1)$
• 2012: Thomas Ernst: A Comprehensive Treatment of q-Calculus ... (previous) ... (next): ($6.3$)

$q$-calculus

$q$-Binomial

$q$-Gamma

$q$-factorial

$q$-Pochhammer