Difference between revisions of "Q-Pochhammer"
From specialfunctionswiki
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| − | $$(a;q)_n=\ | + | The $q$-Pochhammer symbol $(a;q)_n$ is defined for $n=0$ by $(a;q)_0=1$, for $n=1,2,3,\ldots$ by the formula |
| − | + | $$(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).$$ | ||
| − | |||
=Properties= | =Properties= | ||
| + | [[Series for q-Pochhammer]]<br /> | ||
[[Relationship between q-factorial and q-pochhammer]]<br /> | [[Relationship between q-factorial and q-pochhammer]]<br /> | ||
[[Relationship between Euler phi and q-Pochhammer]]<br /> | [[Relationship between Euler phi and q-Pochhammer]]<br /> | ||
Revision as of 19:31, 15 December 2016
The $q$-Pochhammer symbol $(a;q)_n$ is defined for $n=0$ by $(a;q)_0=1$, for $n=1,2,3,\ldots$ by the formula $$(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
Series for q-Pochhammer
Relationship between q-factorial and q-pochhammer
Relationship between Euler phi and q-Pochhammer
