Difference between revisions of "Q-Binomial"
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| − | $${n \brack | + | The $q$-Binomial function is |
| − | where $(q;q)_k$ is the [[q-Pochhammer symbol]]. | + | $${n \brack k}_q = \dfrac{[n]_q!}{[n-k]_q![k]_q!} = \dfrac{(q;q)_n}{(q;q)_k (q;q)_{n-k}},$$ |
| + | where $[n]_q!$ denotes the [[q-factorial|$q$-factorial]] $(q;q)_k$ is the [[q-Pochhammer symbol|$q$-Pochhammer symbol]]. | ||
=Properties= | =Properties= | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
<strong>Theorem:</strong> For $|x|<1,|q|<1$, | <strong>Theorem:</strong> For $|x|<1,|q|<1$, | ||
| − | $$\displaystyle\sum_{k=0}^{\infty} \dfrac{(a;q)_k}{(q;q)_k} x^k = \dfrac{(ax;q)_{\infty}}{(x;q)_{\infty}} | + | $$\displaystyle\sum_{k=0}^{\infty} \dfrac{(a;q)_k}{(q;q)_k} x^k = \dfrac{(ax;q)_{\infty}}{(x;q)_{\infty}}.$$ |
| − | |||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> proof goes here █ | <strong>Proof:</strong> proof goes here █ | ||
</div> | </div> | ||
</div> | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
| + | <strong>Corollary:</strong> | ||
| + | * $\displaystyle\sum_{k=0}^{\infty} \dfrac{x^k}{(q;q)_k} = \dfrac{1}{(x;q)_{\infty}}; |x|<1,|q|<1$ | ||
| + | * $\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kq^{k \choose 2}x^k}{(q;q)_k} =(x;q)_{\infty} ; |q|<1$ | ||
| + | * $\displaystyle\sum_{k=0}^N {N \brack k}_q (-1)^k q^{k \choose 2} x^k = (x;q)_N = (1-x)\ldots(1-xq^{N-1})$ | ||
| + | * $\displaystyle\sum_{k=0}^{\infty} {{N+k-1} \brack k}_q x^k = \dfrac{1}{(x;q)_N} = \dfrac{1}{(1-x)\ldots(1-xq^{N-1})} ;|x|<1$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> proof goes here █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | =References= | ||
| + | *Special Functions - G. Andrews, R. Askey, R. Roy | ||
| + | |||
| + | {{:q-calculus footer}} | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 18:56, 24 May 2016
The $q$-Binomial function is $${n \brack k}_q = \dfrac{[n]_q!}{[n-k]_q![k]_q!} = \dfrac{(q;q)_n}{(q;q)_k (q;q)_{n-k}},$$ where $[n]_q!$ denotes the $q$-factorial $(q;q)_k$ is the $q$-Pochhammer symbol.
Properties
Theorem: For $|x|<1,|q|<1$, $$\displaystyle\sum_{k=0}^{\infty} \dfrac{(a;q)_k}{(q;q)_k} x^k = \dfrac{(ax;q)_{\infty}}{(x;q)_{\infty}}.$$
Proof: proof goes here █
Corollary:
- $\displaystyle\sum_{k=0}^{\infty} \dfrac{x^k}{(q;q)_k} = \dfrac{1}{(x;q)_{\infty}}; |x|<1,|q|<1$
- $\displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kq^{k \choose 2}x^k}{(q;q)_k} =(x;q)_{\infty} ; |q|<1$
- $\displaystyle\sum_{k=0}^N {N \brack k}_q (-1)^k q^{k \choose 2} x^k = (x;q)_N = (1-x)\ldots(1-xq^{N-1})$
- $\displaystyle\sum_{k=0}^{\infty} {{N+k-1} \brack k}_q x^k = \dfrac{1}{(x;q)_N} = \dfrac{1}{(1-x)\ldots(1-xq^{N-1})} ;|x|<1$
Proof: proof goes here █
References
- Special Functions - G. Andrews, R. Askey, R. Roy
