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.

Filter by
Sorted by
Tagged with
3
votes
1answer
614 views

Looking for the limit of a sum

Looking for a limit, with $1<\alpha\leq 2$, $\sigma>0$: $$\lim_{p\to \infty } \, \sum _{k=1}^p \left( \frac{1}{2 \pi k!}\left(1+i \tan \left(\frac{\pi \alpha }{2}\right)\right)^{1/\alpha } \...
1
vote
1answer
334 views

Infinite series with a rising factorial?

Is there a closed-form expression for the infinite series $\sum_{i=0}^\infty (-\pi)^i\alpha^{(i)}$ For known $\pi,\alpha\in [0,1)$ where $\alpha^{(i)}$ is the rising factorial or Pochhammer symbol $\...
0
votes
1answer
31 views

Is $\frac{(\alpha)^n (\beta)^n} {(\delta)^n} > \frac{(\alpha+1)^n (\beta+1)^n} {(\delta+1)^n}$ for any $n$ ?(in this specific case)

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} {(\...
1
vote
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 ...
4
votes
1answer
147 views

Product identity for $n^n$

I 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}&&\...
2
votes
3answers
332 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? ...
2
votes
1answer
85 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 $x \ne 0$ ...
0
votes
1answer
760 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 ...
4
votes
1answer
222 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. $(a)_{n}=a\cdot(a+1)\cdots(a+n-1)$? ...
4
votes
3answers
532 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 ...
6
votes
1answer
306 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 $0<q<1$....
4
votes
2answers
706 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 ...
3
votes
1answer
845 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 ...
2
votes
2answers
170 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}}{...
1
vote
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 $...
4
votes
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 $\sum_{k=0}^n a^k n^{\underline k}$?
25
votes
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}^\...
1
vote
1answer
72 views

Definition domains of the pochhammer symbols?

What are the definition domains for $n$ and $x$ that gives $x^{(n)}$ (upper pochhammer symbol) and $(x)_n$ (lower pochhammer symbol) in $\mathbb{R}$ ?
4
votes
0answers
148 views

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 $r_{1}$ ...