All Questions
Tagged with closed-form gamma-function
109
questions
60
votes
1
answer
6k
views
Is it possible to simplify $\frac{\Gamma\left(\frac{1}{10}\right)}{\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$?
Is it possible to simplify this expression?
$$\frac{\displaystyle\Gamma\left(\frac{1}{10}\right)}{\displaystyle\Gamma\left(\frac{2}{15}\right)\ \Gamma\left(\frac{7}{15}\right)}$$
Is there a systematic ...
- 2,602
50
votes
5
answers
1k
views
Closed form of $\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$
Is there a closed form of the following infinite product?
$$\prod_{n=1}^\infty\sqrt[2^n]{\frac{\Gamma(2^n+\frac{1}{2})}{\Gamma(2^n)}}$$
- 13.2k
48
votes
1
answer
1k
views
How to evaluate double limit of multifactorial $\lim\limits_{k\to\infty}\lim\limits_{n\to 0} \sqrt[n]{n\underbrace{!!!!\cdots!}_{k\,\text{times}}}$
Define the multifactorial function $$n!^{(k)}=n(n-k)(n-2k)\cdots$$ where the product extends to the least positive integer of $n$ modulo $k$. In this answer, I derived one of several analytic ...
- 27.1k
34
votes
2
answers
2k
views
Closed form for $\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$
I need to find a closed form for these nested definite integrals:
$$I=\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$$
The inner integral can be ...
- 47.3k
33
votes
3
answers
2k
views
How to prove $\int_0^\infty J_\nu(x)^3dx\stackrel?=\frac{\Gamma(1/6)\ \Gamma(1/6+\nu/2)}{2^{5/3}\ 3^{1/2}\ \pi^{3/2}\ \Gamma(5/6+\nu/2)}$?
I am interested in finding a general formula for the following integral:
$$\int_0^\infty J_\nu(x)^3dx,\tag1$$
where $J_\nu(x)$ is the Bessel function of the first kind:
$$J_\nu(x)=\sum _{n=0}^\infty\...
- 47.3k
33
votes
1
answer
1k
views
Closed form for $\sum_{n=1}^\infty\frac{\psi(n+\frac{5}{4})}{(1+2n)(1+4n)^2}$
This question came up in the process of finding solution to another problem. Eventually, the problem was solved avoiding calculation of this sum, but it looks quite interesting on its own. Is there a ...
- 13.2k
28
votes
3
answers
1k
views
Interesting closed form for $\int_0^{\frac{\pi}{2}}\frac{1}{\left(\frac{1}{3}+\sin^2{\theta}\right)^{\frac{1}{3}}}\;d\theta$
Some time ago I used a formal approach to derive the following identity:
$$\int_0^{\frac{\pi}{2}}\frac{1}{\left(\frac{1}{3}+\sin^2{\theta}\right)^{\frac{1}{3}}}\;d\theta=\frac{3^{\frac{1}{12}}\pi\...
- 3,343
26
votes
3
answers
1k
views
Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$
Please help me to find a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
- 2,575
26
votes
1
answer
939
views
Is $K\left(\frac{\sqrt{2-\sqrt3}}2\right)\stackrel?=\frac{\Gamma\left(\frac16\right)\Gamma\left(\frac13\right)}{4\ \sqrt[4]3\ \sqrt\pi}$
Working on this conjecture, I found its corollary, which is also supported by numeric calculations up to at least $10^5$ decimal digits:
$$K\left(\frac{\sqrt{2-\sqrt3}}2\right)\stackrel?=\frac{\Gamma\...
- 47.3k
25
votes
1
answer
1k
views
Strange closed forms for hypergeometric functions
So in the process of trying to find a derivation for this answer, the following interesting equalities arose (one can check with Wolfram Alpha/Mathematica):
$$\frac{8\sqrt{2}G^4}{5\pi^2} \left(\left(...
- 1,827
23
votes
3
answers
1k
views
Integral $\int_{-\infty}^\infty\frac{\Gamma(x)\,\sin(\pi x)}{\Gamma\left(x+a\right)}\,dx$
I would like to evaluate this integral:
$$\mathcal F(a)=\int_{-\infty}^\infty\frac{\Gamma(x)\,\sin(\pi x)}{\Gamma\left(x+a\right)}\,dx,\quad a>0.\tag1$$
For all $a>0$ the integrand is a smooth ...
- 1,271
22
votes
2
answers
2k
views
Closed form (or an ODE) for the integral $\int_0^\infty \frac{1+z^2}{1+z^4} \frac{z^p}{1+z^{2p}} dz$
Is there a closed form for: $$I(p)=\int_0^\infty \frac{1+z^2}{1+z^4} \frac{z^p}{1+z^{2p}} dz$$
The integral has a number of nice properties:
$$I(p)=I(-p)$$
$$I(p)=2\int_0^1 \frac{1+z^2}{1+z^4} \...
- 31.7k
21
votes
3
answers
1k
views
On the general form of the family $\sum_{n=1}^\infty \frac{n^{k}}{e^{2n\pi}-1} $
I. $k=4n+3.\;$ From this post, one knows that $$\sum_{n=1}^\infty \frac{n^{3}}{e^{2n\pi}-1} = \frac{\Gamma\big(\tfrac{1}{4}\big)^8}{2^{10}\cdot5\,\pi^6}-\frac{1}{240}$$ and a Mathematica session ...
- 54.9k
18
votes
1
answer
400
views
Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n}$
Is there a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
This is a ...
- 2,575
17
votes
1
answer
1k
views
What is $\mathcal{R}$?
First of all, I am asking this question entirely out of curiosity. It basically randomly popped out of my mind.
So I am asking for the value of an infinite series.
Let's call it, $\mathcal{R}=\sum_{n=...
- 2,469