Revision as of 19:33, 15 December 2016

The $q$-Pochhammer symbol $(a;q)_n$ is defined for $n=0$ by $(a;q)_0=1$, for $n=1,2,3,\ldots$ by the formula $$(a;q)_n= \displaystyle\prod_{k=0}^{n-1} (1-aq^k),$$ and $$(a;q)_{-n}=\dfrac{1} {(aq^{-n};q)_n} =\dfrac{1} {(1-aq^{-n})\ldots(1-aq^{-1})} = \dfrac{q^{\frac{n(n+1)}{2}}(-1)^n}{a^n (\frac{q}{a};q)_n}.$$ The notation $(a;q)_{\infty}$ is often encountered and refers to the limit $$\displaystyle\lim_{n\rightarrow \infty} (a;q)_n=\displaystyle\prod_{k=0}^{\infty} (1-aq^k).$$

Properties

q-Pochhammer as sum of q-binomial coefficients
Relationship between q-factorial and q-pochhammer
Relationship between Euler phi and q-Pochhammer

$q$-Binomial

$q$-Gamma

$q$-factorial

$q$-Pochhammer