Q-factorial

The $q$-Factorial is defined for a non-negative integer $k$ by $$[n]_q! = [1]_q [2]_q \ldots [n]_q=1(1+q)(1+q+q^2)\ldots(1+q+\ldots+q^{n-1})=\dfrac{(q;q)_n}{(1-q)^n},$$ where $[k]_q$ denotes a $q$-number and $(q;q)_k$ is the q-Pochhammer symbol.

Properties

Theorem

The following formula holds: $$D_q(z^n)=[n]_q z^{n-1},$$ where $D_q$ denotes the $q$-derivative and $[n]_q$ denotes the $q$-number.

References

• D.S. McAnally: q-exponential and q-gamma functions. I. q-exponential functions (1994)... (previous)... (next) $(2.2)$Relationship between q-factorial and q-pochhammer

$q$-calculus

$q$-Binomial

$q$-Gamma

$q$-factorial

$q$-Pochhammer