# 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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### $n$-th derivative of $x^\alpha$ where $\alpha = m + 1/2$

It is well-known that, for any real $\alpha$ and nonnegative integer $n$ $$\frac{d^n x^\alpha}{dx^n} = \alpha(\alpha-1)\cdots(\alpha - n + 1) x^{\alpha - n}$$ I just found out that the coefficient ...
1answer
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### Integral involving the Pochhammer symbol

I am wondering do we know any integral idenitiies involving the Pochhammaer symbol? More specficically, if $(x)_n$ is the Pochhammer symbole, do we know any functions f and g make the following the ...
1answer
133 views

### The behaviour of $\operatorname{Im}(!n)$

What's going on with the behaviour of the subfactorial's imaginary part? Background: Out of curiosity I tried to construct some recurrence relations using the Pochhammer symbol and out of those came ...
0answers
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0answers
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### Can this sum over the q-Pochhammer symbol be simplified?

While considering the problem of the expected value of a dice fixing strategy on a two-sided die that comes up as $1$ with a probability of $\alpha$ and $0$ otherwise. I was studying the strategy ...
0answers
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### show $\sum_{j=0}^n (-1)^j {n \brack j}_q =0$ for n odd

I would like to show $\sum_{j=0}^n (-1)^j {n \brack j}_q =0$ for n odd, or preferably even more generally that $\sum_{j=0}^n (-1)^j {n \brack j }_q =\frac{1}{2}((-1)^n+1)(q;q)_{\frac{n}{2}}$. Using ...
0answers
46 views

### Product of a modified/generalized geometric progression

What is the solution to the following? It's sort of a modified/generalized geometric progression (or is there a known name for this kind?), $$\lim_{N\to\infty}\prod_{n=0}^{N}(1+a^n), a\in(0,1)$$
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 $r_{1}$ ...
5answers
232 views

### Coefficient of $x^{50}$ in the expansion of $\prod_{n=1}^{52}{(x+n)}$

Find the coefficient of $x^{50}$ in the expansion of $$\prod_{n=1}^{52}{(x+n)}$$ I can't find a way out.
1answer
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### Prove that ${}_2F_1(0,b;c;z)=1$

I do not know how I could prove that ${}_2F_1(0,\beta;\gamma;t)=1$ because when I apply the definition I get $0$, namely.. $$\sum_{n=0}^{\infty}\frac{(0)_n(\beta)_n}{n!(\gamma)_n}t^n=0$$ someone ...
1answer
144 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}&&\...
2answers
225 views

### Mathematical expressions for binomial coefficient and Pochhammer’s Symbol with negative values

I have two questions regarding the binomial coefficient and Pochhammer’s Symbol when they contain negative value; In the following example $\sum\limits_{k=0}^{-n} \binom{-n}{k} \left(a\right)_{-n}$. ...
2answers
604 views

### Integral of binomial coefficients

Let the integral in question be given by \begin{align} f_{n}(x) = \int_{1}^{x} \binom{t-1}{n} \, dt. \end{align} The integral can also be seen in the form \begin{align} f_{n}(x) = \frac{1}{n!} \, \...