# Questions tagged [pochhammer-symbol]

The Pochhammer symbol is the notation used for rising and falling factorials. The $q$-Pochhammer symbol is the $q$-analog.

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Let $\alpha=(K-1)a$, $\beta=K$ and $\delta=Ka$, where $K>a\ge 1$ ($\delta>\alpha>\beta$). Can we claim that $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} {(\... 0answers 34 views ### is there anyone able to develop this series in order to get the following equality?$\sum_{i=1}^\infty (1-\alpha)_{(i-1)}*\frac{\varepsilon^i}{i!}$=$\frac{1-(1-\varepsilon)^{\alpha}}{\alpha}$where$(1-\alpha)_{(i-1)}$is the Pochammer symbol or rising\ascending factorial. Can ... 1answer 144 views ### Product identity for$n^nI came across the rather nice identity \begin{align} &&\frac{(-n)^{n-1} \Gamma (n+1)}{(1-n)_{n-1}}&&\tag{1}&\\ \\ &=&\prod _{k=1}^{n-1} \frac{(k+1) n^2}{n^2-k n}&&\... 3answers 326 views ### Another question about ratios of Pochhammer symbols My question is similar to this question. Can $$\frac{(11/6)_n (7/6)_n (3/2)_n}{(3)_n}$$ be expressed 'nicely' in terms of factorials just like(1/6)_n (1/2)_n (5/6)_n$in the aforementioned question? ... 1answer 84 views ### Strange claim by WA involving nth derivative Playing a bit around with WA i found this Namely: $$\frac {d^n}{d^nx} \left(\frac x{f(x)}\right)^{n+1}=x\left(\frac 1{f(x)}\right)^{n+1}(2)_n$$ For$n\in\mathbb{N_0}$and$n+1\ne x$and$x \ne 0$... 1answer 722 views ### Understanding relation between Product and Summation Notation So I am given the following:$n = \sum_{i=1}^{k}m_{i}$I am also given$x = \sum_{i=1}^{k}log(m_{i}) = log\prod_{i=1}^{k}m_{i}$I was only given the first part, however I believe that is a ... 1answer 218 views ### Coefficients in Pochhammer Expansion Can anyone tell me if there is a formula for finding the coefficient of$x^3$in the expansion of$(3x+5)_{6}$, where$(a)_n$denotes the Pochhammer symbol, i.e.$(a)_{n}=a\cdot(a+1)\cdots(a+n-1)$? ... 3answers 523 views ### An identity involving the Pochhammer symbol I need help proving the following identity: $$\frac{(6n)!}{(3n)!} = 1728^n \left(\frac{1}{6}\right)_n \left(\frac{1}{2}\right)_n \left(\frac{5}{6}\right)_n.$$ Here, $$(a)_n = a(a + 1)(a + 2) \cdots (a ... 1answer 304 views ### Combinatorial Identity \sum_{k=1}^n (-1)^{k-1} \cdot q^{\frac{k(k-1)}{2}} \cdot \frac{\prod_{i=n-k+1}^n(1-q^i)}{\prod_{i=1}^k(1-q^i)} = 1 I have to validate the following identity which is defined:$$ \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 $$where 0<q<1.... 2answers 689 views ### Unimodality of q-binomial coefficients The q-Pochhammer symbol [n]_q! is defined as$$[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})$$It can be easily shown that [n]_q! (function ... 1answer 837 views ### Identity using q-Pochhammer symbols Prove -$$∑_{n=0}^{∞} \frac{(a;q)_n}{(q;q)_n} q^{n\choose 2} q^n={(−q;q)_∞}{(aq;q^2)_∞}.$$where (a;q) are the q-Pochhammer symbols. I know that the RHS is the product of generating functions of ... 2answers 169 views ### Identity involving the rising factorial I am reading a book about hypergeometric functions and in a proof of a transformation they use the supposedly obvious fact$$ \displaystyle\frac{(c-a-b)_{n-r}}{(n-r)!} = \frac{(c-a-b)_{n} (-n)_{r}}{... 1answer 94 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$...
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 $\sum_{k=0}^n a^k n^{\underline k}$?