Mathematics Stack Exchange
Mathematics Stack Exchange

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.

The Pochhammer symbol is the notation used for the rising factorial:

$$(x)^n=x(x+1)\dots(x+n-1)$$

and the falling factorial:

$$(x)_n=x(x-1)\dots(x-n+1)$$

The $q$-Pochhammer symbol, also known as the $q$-shifted factorial, is defined by:

$$ (a;q)_n = \prod\limits_{k=0}^{n-1} (1-aq^k) $$

The Euler function $\phi(q)$ can be written as $(q;q)_\infty$.

126 questions
58
votes
2 answers

Is the new series for a Big (or even Medium) Deal?

There's been some oohing and ahhing in the science press recently over the discovery of a new formula for $\pi$ obtained as a side effect of computing amplitudes in string theory: $$\pi=4+\sum_{n=1}^\infty…
Steven Stadnicki
  • 53,147
26
votes
5 answers

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}^\infty\frac{2^j j!}{(j)_j} = \frac{2^2 2!}{(2)_2} +…
JohnWO
  • 2,119
25
votes
1 answer

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}\sum_{n=1}^\infty…
Carolus
  • 3,369
15
votes
2 answers

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!} \, \int_{1}^{x} \frac{\Gamma(t) \, dt}{\Gamma(t-n)} =…
Leucippus
  • 27,174
13
votes
2 answers

Summation of $\sum_{n=0}^{\infty}a^nq^{n^2}$

I am trying to find the result for the sum of the form $\sum_{n=0}^{\infty}a^nq^{n^2}$. The special case for $a=1$ is easily given by $\vartheta(0,q)$, where $\vartheta(z,q)$ is the third Jacobi Theta function. So, whatever the answer is, it must…
ck1987pd
  • 1,129
10
votes
1 answer

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 some subfactorials. For…
Carolus
  • 3,369
8
votes
2 answers

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 of indeterminate $q$) is symmetric and unimodal,…
UnadulteratedImagination
  • 822
7
votes
1 answer

Asymptotic expansion of q-Pochhammer symbol near q = 1

I'd like to understand the asymptotics of the q-Pochhammer symbol $(a;q)_\infty$ as $q \to 1^-$ with $a$ complex, where $$(a;q)_\infty = \prod_{n = 0}^\infty (1- aq^n).$$ More specifically, I'm actually just interested in the limiting behavior as…
Sebastian
  • 642
6
votes
1 answer

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
Tuan
  • 75
6
votes
1 answer

Is it true that $\sum\limits_{n\ge1}\binom{n+\frac1{4n}-1}n=\frac37$?

$$ \mbox{Is this closed form true ?:}\quad \sum_{n \geq 1}{n + 1/\left(4n\right) - 1 \choose n} =\frac37 $$ The series arises upon taking $\lambda=0$ in the identity from another post $$ \pi = 4+\sum_{n=1}^\infty…
TheSimpliFire
  • 28,020
6
votes
2 answers

Problematic limit $\epsilon \to 0 $ for combination of hypergeometric ${_2}F_2$ functions

In an earlier question, the integral $$I_n(c)=\int_0^\infty x^n (1+x)^n e^{-n c x^2} dx$$ was considered with particular focus on its behavior for positive integer $n$. In trying to analyze this, it appears that Mathematica runs into issues in…
Semiclassical
  • 18,592
6
votes
1 answer

Identity involving double sum with factorials

In the course of a calculation, I have met a complicated identity, which I want to prove. Let $m>0$ and $0<\ell
Marcel
  • 1,613
6
votes
1 answer

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-trivial integer solutions $(n,k)$? $$(n)_k=n!$$ Among the…
Yuriy S
  • 32,728
5
votes
0 answers

Using Ramanujan-type series for $1/\pi^m$ to find formulas for $\zeta(2),\, \zeta(3),\, \zeta(4)$?

As described by Guillera in "Ramanujan Series with a Shift", one nice thing about Ramanujan-type $1/\pi^m$ formulas is by "shifting" them, they can yield a second value which may also have a closed-form. For $m>1$, only 12 non-trivial ones are known…
Tito Piezas III
  • 60,745
5
votes
0 answers

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 where one fixes every die if they are all $1$'s and…
Milo Brandt
  • 61,938
1
2 3
…
8 9