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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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}^\...
21
votes
1answer
539 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}\...
11
votes
2answers
601 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!} \, \...
6
votes
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 $0<q<1$....
6
votes
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 ...
6
votes
1answer
91 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-...
4
votes
3answers
522 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 ...
4
votes
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 ...
4
votes
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. $(a)_{n}=a\cdot(a+1)\cdots(a+n-1)$? ...
4
votes
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}&&\...
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
votes
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.
3
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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 ...
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 } \...
3
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1answer
57 views

$n$-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 ...
3
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0answers
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
votes
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}}{...
2
votes
2answers
85 views

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}-(...
2
votes
1answer
206 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 ...
2
votes
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? ...
2
votes
1answer
329 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. ...
2
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1answer
138 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+...
2
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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 $x \ne 0$ ...
2
votes
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
votes
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
votes
2answers
132 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
votes
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
vote
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 ...
1
vote
2answers
54 views

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 ...
1
vote
1answer
118 views

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{(...
1
vote
2answers
58 views

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(...
1
vote
1answer
49 views

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 ...
1
vote
1answer
128 views

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 $a=1,2,3,\dots$, $b=2,3,4,\dots$, $c>0$...
1
vote
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 $\sum\limits_{k=0}^{-n} \binom{-n}{k} \left(a\right)_{-n}$. ...
1
vote
1answer
238 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, ...
1
vote
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 $...
1
vote
1answer
71 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 $\mathbb{R}$ ?
1
vote
1answer
31 views

To obtain a closed form for the series related to special functions.

I am learning the properties of special functions particularly, hypergeometric functions. I got the following series form: $$f(z)=\Gamma(3)\left[\frac{z}{\Gamma(3)}+\frac{(\gamma)_1}{\Gamma(4)}\frac{z^...
1
vote
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 ...
1
vote
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
vote
2answers
232 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} ...
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
vote
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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0answers
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 $\...