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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21
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1answer
541 views

An interesting formula for $\pi$

Looking through some old notebooks I found this monster of a formula: For any integer $r>1$, we have $$\pi=(-1)^{\left\lfloor\frac{r}{2}\right\rfloor-\left\lfloor\frac{2r-1}{4}\right\rfloor}\...
4
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0answers
300 views

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 ...
4
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0answers
186 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 $\sum_{k=0}^n a^k n^{\underline k}$?
4
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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 $r_{1}$ ...
3
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181 views

Is this expansion related to Pochhammer's symbol, the gamma function, and beta function valid for $a<0$ and $a\neq 0,-1,-2,\ldots$?

If I start with the definition of the beta function $$B(a,b) = \int_0^1 t^{a-1} (1-t)^{b-1} \operatorname{d}t = \frac{\Gamma(a) \Gamma(b)}{\Gamma(a+b)}$$ that is valid for $\mathcal{R}(a) >0$ and $\...
2
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0answers
48 views

Expansion of the falling factorial

Let $(x)_n=x(x-1)\ldots (x-(n-1))$ be the falling factorial. For small $n$ I have found $$ (xy)_1=(x)_1 (y)_1,\\ (xy)_2=x (x)_1 (y)_2+(x)_2 (y)_1,\\ (xy)_3=x^2 (x)_1 (y)_3+3 x (x)_2(y)_2+(x)_3(y)_1 $...
2
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0answers
79 views

distribution of partitions of N into m distinct parts bounded by L

For my application, the expression I'm interested in is: \begin{equation} \prod_{j=1}^{L}(1+xq^{j})=\sum_{k=0}^{L}x^{k}q^{\frac{k(k+1)}{2}}\sum_{n=0}^{k(L-k)}q_{L\geq}(n+\frac{k(k+1)}{2},k)q^{n}, \...
2
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0answers
61 views

Generalization of factorial powers of custom step to complex power, it is possible?

The falling and rising factorial are defined by $$z^\underline n:=\prod_{k=0}^{n-1}(z-k),\quad z^\overline n:=\prod_{k=0}^{n-1}(z+k),\quad z\in\Bbb C,n\in\Bbb N_{\ge 0}\tag{1}$$ In first place $(1)$ ...
2
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2answers
133 views

Showing a finite sum involving Gamma functions adds to zero

In the process of proving the Wrongskian identity for the Bessel function $J_\nu(x)J_{-\nu}'(x)-J_\nu'(x)J_{-\nu}(x)=-\frac{2\sin(\pi \nu)}{\pi x}$ (where the primes are differentiation by $x$), I ...
2
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0answers
163 views

Expected number of partitions in Pitman-Yor process

I am reading two articles on this and am trying to reach from the exact formula of this (page 19), to the approximate formula of this (section 3.2). Here are the exact and approximated forms using a ...
2
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0answers
50 views

Prove the asymptote $\Gamma \left(x+\frac{1}{\Gamma(x-y)} \right)/ \Gamma \left(\frac{1}{\Gamma(x-y)} \right) \asymp (x-2)^y$ as $x \to \infty$

I noticed it first for Pochhammer symbols for integers: $$\left( \frac{1}{n!} \right)_n \asymp \frac{1}{n} \quad \text{as} \quad n \to \infty$$ $$\left( \frac{1}{n!} \right)_{n+1} \asymp 1 \quad \...
2
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0answers
42 views

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 ...
1
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0answers
50 views

Summing nearly successive falling factorials

Define the functions $Q_b(a) = \prod_{k=1}^{2b} \big( a +1 - \frac{k}{2}\big)= (a+\frac{1}{2})a(a-\frac{1}{2}) \dotsb (a+1-b)$, and consider the sum $$\sum_{a=0}^{r-2} Q_b(a+m)\,.$$ This sum is ...
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0answers
111 views

Solve ODE $y''(x) - iy'(x)-\alpha y(x)/x^{2} = 0$

I want to know how to solve the above equation. x is defined in [0,1] and $\alpha$ is a constant. Wolfram Alpha already gives me a solution. I'm trying to solve using Frobenius method, assuming a ...
1
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0answers
56 views

The integral $\int_0^{\frac{1}{2}}\frac{x^{k-1}}{1-x^{2^k}}dx$ and how simplify the Pochhmammer symbol in related series

Inspired in the shape of useful integrals to compute $\pi$ (see *), I've consider for each integer $k\geq 1$ $$\int_0^{\frac{1}{2}}\frac{x^{k-1}}{1-x^{2^k}}dx=\int_0^{\frac{1}{2}}x^{k-1}\sum_{n=0}^\...
1
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0answers
86 views

Falling Factorial Notation

If $(x)_{n}$ refers to $$x(x-1)\cdots(x-n+1)$$ then what does $(xy;x)_{n}$ refer to? Is it $$xy(xy-1)\ldots(xy-n+1)?$$ Thanks. The notation in question is used on page two of this paper.
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97 views

Q Pochammer Symbol Product Identities

Consider the expression $$G(x,a) = \frac{1}{((1-a)x;a)_{\infty}}$$ Based on: Infinite sum involving ascending powers It follows that in the limit as $a \rightarrow 1$ $$\frac{1}{((1-a)x;a)_{\...
1
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1answer
328 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 $\...
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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 ...
0
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0answers
49 views

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 ...
0
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1answer
41 views

What is the limit of the series (summation) of the q-Pochhammer symbol or the ~q-Pochhammer symbol?

I am interested in knowing if the following series converges or not: \begin{equation} \sum_{n=1}^{\infty} \prod_{i=1}^n \left(1-e^{-\sqrt{i}} \right) \qquad Expression \; 1 \end{equation} If that is ...
0
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0answers
24 views

Simplifying terms with Pochhammer symbol

Is the simplification from line 1 to 2 correct? BTW, $(a)_{k}$ is the usual Pochhammer symbol. $\rho=\frac{b_0}{1-q}\frac{x^a}{a}\sum_{k=0}^\infty(-1)^k\frac{(b-1)_k}{k!}\frac{(a)_k}{(a+1)_k} x^k$ $=...
0
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0answers
68 views

Binomial Coefficients $ {\frac{-1}{2} \choose n}$

i am trying to solve Exercises from Martin Aigner A Copurse in Enummarations, i am having problems with this one: I got the first part(well you open the factorials by definition and put the 2's in ...
0
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1answer
201 views

Definite integral of exponential, power and Bessel function : a hypergeometric function?

I am studying the following integral: $\int_0^T \frac{e^{-x}}{x}I_n(\alpha x)dx$ I have discovered some things about it but I'm not yet satisfied with it. The first thing I tried is using the ...
0
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1answer
94 views

Pochhammer symbol finite summatory

I need some help in showing that in product among $n$ lower triangular matrices, the number of addends to be summed in order to obtain the value of the elements $(i, j)$ is: $\frac{<n>_{i-j}}{(i-...
0
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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?), $$\lim_{N\to\infty}\prod_{n=0}^{N}(1+a^n), a\in(0,1)$$
-2
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1answer
95 views

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 ...