Questions tagged [pochhammer-symbol]

The Pochhammer symbol is notation used for both rising and falling factorials, e.g. in defining basic hypergeometric series and related special functions. This tag is also appropriate for questions about the $q$-Pochhammer symbol, which plays a similar role in defining $q$-hypergeometric series, etc.

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Combinatorial Identity $\sum_{k=1}^n (-1)^{k-1} \cdot q^{\frac{k(k-1)}{2}} \cdot \frac{\prod_{i=n-k+1}^n(1-q^i)}{\prod_{i=1}^k(1-q^i)} = 1$

I have to validate the following identity which is defined: $$\sum_{k=1}^n (-1)^{k-1} \cdot q^{\frac{k(k-1)}{2}} \cdot \frac{\prod_{i=n-k+1}^n(1-q^i)}{\prod_{i=1}^k(1-q^i)} = 1$$ where $0<q<1$....
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Unimodality of q-binomial coefficients

The q-Pochhammer symbol $[n]_q!$ is defined as $$[n]_q! = \frac{(1-q^n)(1-q^{n-1})\cdots(1-q)}{(1-q)^n} = (1+q) (1+q+q^2) \cdots (1+q+\cdots+q^{n-1})$$ It can be easily shown that $[n]_q!$ (function ...
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Identity using q-Pochhammer symbols

Prove - $$∑_{n=0}^{∞} \frac{(a;q)_n}{(q;q)_n} q^{n\choose 2} q^n={(−q;q)_∞}{(aq;q^2)_∞}.$$ where $(a;q)$ are the q-Pochhammer symbols. I know that the RHS is the product of generating functions of ...
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I am reading a book about hypergeometric functions and in a proof of a transformation they use the supposedly obvious fact $$\displaystyle\frac{(c-a-b)_{n-r}}{(n-r)!} = \frac{(c-a-b)_{n} (-n)_{r}}{... 1answer 102 views The multinomial formula as three Pochhammer rising factorials I need to describe:$${n \choose k,0,l,0,m}$$as three rising factorials. How can I do this? As far as I know I can delete zero's, so it would be:$${n \choose k,l,m}=\frac{n!}{k!l!m!},$$where ... 0answers 187 views Binomial-like sum involving falling factorials We know that \sum_{k=0}^n a^k \frac{n^{\underline k}}{k!} = (1+a)^n. Is there a known (preferably closed) form for \sum_{k=0}^n a^k n^{\underline k}? 4answers 1k views Trying to prove that \sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi How could one prove that:$$\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$$This is about as far as I got:$$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies\sum_{j=2}^\...
What are the definition domains for $n$ and $x$ that gives $x^{(n)}$ (upper pochhammer symbol) and $(x)_n$ (lower pochhammer symbol) in $\mathbb{R}$ ?
factoring infinite products of $q$-series with constant term equal to 1
I was thinking about the following infinite product: $$\prod_{n=0}^{\infty} \frac{ae^{-2n}+be^{-n}+c}{c}$$ The right way of generalizing it is to think in terms of $q$-Pochhammer symbols. If $r_{1}$ ...