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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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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 ...
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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 ...
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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}\...
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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 ...
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Simplifying terms with Pochhammer symbol

Is the simplification from line 1 to 2 correct? BTW, $(a)_{k}$ is the usual Pochhammer symbol. $\rho=\frac{b_0}{1-q}\frac{x^a}{a}\sum_{k=0}^\infty(-1)^k\frac{(b-1)_k}{k!}\frac{(a)_k}{(a+1)_k} x^k$ $=...
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Proof of Identity to Zero of the Sum of a Product of Binomial Coefficients & Pochhammer Numbers

It's well-know that the sum across an entire row of binomial coefficients (of degree, say, $n$) with alternating signs attached is 0; and it can easily be proven by demonstrating that it is the ...
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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 $...
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Confused about Arccosecant and Legendre & Pochhammer Notations

Wolfram Math (http://mathworld.wolfram.com/InverseCosecant.html) had a breakdown of Inverse Cosecant (arcsc), a series that looks almost identical to the inverse Sine (arcsin). It (the series) seemed ...
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106 views

Finite sum of the Pochhammer symbols (falling factorials)

Solving a problem I stuck with a sum $$\sum_{k=1}^{n}k \, a^\underline{k}.$$ With a finite difference trick $\Delta_k(a^\underline{k}) = a^\underline{k} \, (a - k - 1)$, written in a book "Concrete ...
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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 ...
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Product identity: $x^{n+1}-1=\prod_{k=0}^{n}{(x-e^\frac{2ik\pi}{n+1}})$

I recently stumbled upon this equation and frankly, I have no idea where this identity comes from. I tried to plot the two sides of the identity as a function and surprisingly, this equation holds. ...
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Constructing a Hypergeometric Function

I am asked to find values for $a,b$ and $c$ such that $$ \frac{1}{2} ((1+x)^{2\alpha}-(1-x)^{2\alpha}) = 2\alpha x\ _2F_1(a,b;c;x^2)$$ I have attempted the following: $$\frac{1}{2} ((1+x)^{2\alpha}-(...
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Rewriting a binomial coefficient in terms of Pochhammer symbols

I am working with the equation $$ \sum^{2n+1}_{k=0} \binom{2n+1}{k}(x^k -(-x)^k), \ n = 0,1,2,..$$ and want to rewrite it in terms of rising Pochhammer symbols. I am aware of the relation $$ \frac{(...
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Double step Pochhammer symbol?

The usual Pochhammer symbol is defined as $$(x)_n=x(x+1)(x+2)...(x+n)=\frac{\Gamma(x+n)}{\Gamma(x)}$$ I am interested in a generalized Pochhammer-like symbol that produces the following output $$x(...
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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}, \...
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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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Finding the coefficient in the expansion of $\prod\limits_{m=1}^N \left(1-R^mA\right)$

I understand that $$\prod\limits_{m=1}^N \left(1-R^mA\right)$$ is a polynomial in $A$, and so can be written as $\sum\limits_{k=0}^N c_k A^k$ for some coefficients $c_k$. I can't seem to figure out a ...
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Solving finite sum in terms of the generalized hypergeometric function

I have the following sum \begin{equation} \sum_{n=1}^{a-1} \frac{(1-a)_{n}}{(2-b)_{n}}\,\frac{\Gamma(n)}{n!}\left(\frac{d}{c}\right)^{n}, \end{equation} where $a=1,2,3,\dots$, $b=2,3,4,\dots$, $c>0$...
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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 $\...
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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 ...
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1answer
98 views

Integrating a Tricky Infinite Sum with a Rising Factorial

The sum that I need help integrating is as follows: $$\int^{k}_{1}\frac{1}{n}+\frac{1}{n(n+k)}+\frac{1}{n(n+k)(n+2k)}+\frac{1}{n(n+k)(n+2k)(n+3k)}+\ ...$$ I was unable to find information on how to do ...
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317 views

Poles of hypergeometric function $_2F_1$

We consider the hypergeometric function $ _2F_1 [\dfrac{1}{2}(1+k+l+\omega), \dfrac{1}{2}(1+k-l+\omega), 1+k, -r^2]$, and use its expansion as given in [1] in terms of the rising Pochhammer symbols. ...
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197 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 ...
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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 $(1)$ ...
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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 ...
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1answer
199 views

What are the coefficients of q-Pochhammer function $(q^n;q)_{\infty}$

I am trying to figure out the coefficients of q-Pochhammer function for special case $(q^n;q)_{\infty}$. I was trying to calculate this using Jacoby's identities but still no success. EDITED: In ...
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229 views

Simple identity involving q-Pochhammer symbol

I have stumbled upon the following fact, easily confirmed numerically: The $q$-Pochhammer symbol $(a;z)_L$ with $z$ given by the $L$th root of unity, $$ (a;\mathrm{e}^{2\pi i/L})_L = \prod_{n=0}^{L-1} ...
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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 ...
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103 views

Are there other definitions of Jacobi polynomials?

While I am reading "On some dual integral equations occurring in potential problems with axial symmetry" by C. J. Tranter, Quarterly Journal of Mechanics & Applied Math (1950) p. 414, the author ...
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90 views

Is $(7,4)$ the only non-trivial integer solution for $(n)_k=n!$?

I accidentally noticed that: $$(7)_4=7 \cdot 8 \cdot 9 \cdot 10=2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7=7!$$ Here $(n)_k$ is the Pochhammer symbol. I wonder, are there any other non-...
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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 \...
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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-...
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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}^\...
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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 ...
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84 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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Improper integral involving sinc function and Pochhammer symbol

Can anyone please advise me how to integrate expressions of the form $\text{sinc}\,(x) / (1-x)_n$ along the real axis? Using a CAS, one could suggest that $$ n! \int_{-\infty}^\infty \frac{\sin \pi x}...
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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 ...
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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?), $$\lim_{N\to\infty}\prod_{n=0}^{N}(1+a^n), a\in(0,1)$$
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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.
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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 ...
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1answer
133 views

How to derive Such infinite sum representation for Hypergeometric function?

I was reading a paper $[1]$ in which authors claimed that we can simplify below Gauss function to finite series if $m $ and $v$ are positive integers. $$ _2F_{1}(v,m+v;m+1;x)=\psi\sum_{c=0}^{v-1} {v+...
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222 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 $\sum\limits_{k=0}^{-n} \binom{-n}{k} \left(a\right)_{-n}$. ...
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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!} \, \...
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1answer
235 views

Falling power of a sum in terms of falling powers of the terms

I am trying to come up with an expression for $(x+y)^{\underline{n}}$ in terms of $x^{\underline{r}}$ and $y^{\underline{r}}$. I tried for $n=2$ and $n=3$ and it looks like binomial expansion holds, ...
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2answers
44 views

How can I show that $\left(a-n-1\right)!/\left(a-1\right)!=\left(-a\right)!\left(-1\right)^n/\left(-a+n\right)!$?

Is it possible to show that \begin{align}\frac{\left(a-n-1\right)!}{\left(a-1\right)!}\stackrel{?}{=}\frac{\left(-a\right)!\left(-1\right)^n}{\left(-a+n\right)!}\tag{1},\end{align} or, more ...
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0answers
95 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)_{\...
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1answer
613 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 } \...
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1answer
321 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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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} {(\...
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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 ...