All Questions
36
questions
4
votes
0
answers
163
views
Closed form for ${_4F_3}\!\left(\begin{array}c\tfrac12,\tfrac12,\tfrac12,\tfrac12\\\tfrac32,\tfrac32,\tfrac32\end{array}\middle|\tfrac12\right)$
I am trying to find the closed form expression for
$${_4F_3}\!\left(\begin{array}c\tfrac12,\tfrac12,\tfrac12,\tfrac12\\\tfrac32,\tfrac32,\tfrac32\end{array}\middle|\tfrac12\right).$$
I encountered ...
5
votes
2
answers
211
views
Prove $_4 F_3\left(2,\frac32,\frac32,\frac32;\frac52,\frac52,\frac52;1\right)=\frac{27}{16}\left(\pi^2-7\zeta(3)\right) $
How to prove the following result about the generalized hypergeometric function $_4 F_3$?
$$_4 F_3\left(2,\frac32,\frac32,\frac32;\frac52,\frac52,\frac52;1\right)\stackrel{?}=\frac{27}{16}\left(\pi^2-...
- 3,392
8
votes
2
answers
324
views
About the Integral $\int\arcsin\left(\sin^{2}x\right)dx$
$$\int\arcsin\left(\sin^{2}x\right)dx$$
I am not able to find a closed form elementary solution for this, though I have no reason to believe it exists.
But trying out the Definite Integral as follows:...
- 3,280
1
vote
1
answer
123
views
Closed form for $\frac{\int_0^1 Ei(x)^5 \ln(x) dx}{\int_0^1 Ei(x)^3\ln(x) dx}$?
Consider the expression
$$\frac{\int_0^1 Ei(x)^5 \ln(x) dx}{\int_0^1 Ei(x)^3 \ln(x) dx} $$
A) Can we rewrite this with a single integral sign?
B) Do we have a closed form for this expression in terms ...
- 16.4k
44
votes
2
answers
3k
views
What is $\, _4F_3\left(1,1,1,\frac{3}{2};\frac{5}{2},\frac{5}{2},\frac{5}{2};1\right)$?
I have been trying to evaluate the series $$\, _4F_3\left(1,1,1,\frac{3}{2};\frac{5}{2},\frac{5}{2},\frac{5}{2};1\right) = 1.133928715547935...$$ using integration techniques, and I was wondering if ...
- 2,017
6
votes
1
answer
281
views
What is $_3F_2\left(1,\frac32,2;\ \frac43,\frac53;\ \frac4{27}\right)$ as an integral?
This was buried in a rather long question, so I'm asking it separately to give it some air. Define,
$$A_3={_3F_2}\left(1,\frac{\color{blue}1}2,\frac22;\ \frac43,\frac53;\ \frac4{27}\right)$$
$$B_3={...
- 54.9k
16
votes
2
answers
599
views
On $\int_0^1\arctan\,_6F_5\left(\frac17,\frac27,\frac37,\frac47,\frac57,\frac67;\,\frac26,\frac36,\frac46,\frac56,\frac76;\frac{n}{6^6}\,x\right)\,dx$
Reshetnikov gave the remarkable evaluation,
\begin{align}
I&= \int_0^1\arctan{_4F_3}\left(\frac15,\frac25,\frac35,\frac45;\frac24,\frac34,\frac54;\frac{1}{64}\,x\right)\,dx \\
&=\frac{3125}{...
- 54.9k
8
votes
0
answers
143
views
More on $\sum_{n=1}^\infty\frac{(4n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}$
Let,
$$\alpha=2\sqrt[3]{1+\sqrt2}-\frac2{\sqrt[3]{1+\sqrt2}}$$
In this post, it was asked if,
$$\sum_{n=1}^\infty\frac{(4\,n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^...
- 54.9k
6
votes
3
answers
624
views
Finding the value of ${_2F_1}\left(\frac12,\frac12;\ 1;\ 1/9\right)$
I need help finding the value of
$${_2F_1}\left(\frac12,\frac12;\ 1;\ \frac{1}{9} \right)\tag1$$
Not in terms of elliptic functions.
I've tried many methods on this and have reduced it down to many ...
- 4,954
1
vote
1
answer
105
views
Integrate $\int_{0}^{\pi}{-\cos{x}}{_2F_1}\left(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};\cos{^{2}x}\right)\sin{^{1+n}x}\sin{^{2}x}^{\frac{-1-n}{2}}$
May I expect the closed-form of this integral?
$$\int_{0}^{\pi}{{_2F_1}\left(\left.\begin{array}{cc}\frac{1}{2}&\frac{-n+1}{2}\\&\frac{3}{2}\end{array}\right|\cos^2(x)\right)(-\cos{x})(\sin{...
- 13
7
votes
1
answer
329
views
Hypergeometric function values and the Baxter constant
While I was working on this question by @Vladimir Reshetnikov, I've found the following relations between Gaussian hypergeometric function values and the Baxter constant:
$$\begin{align}{_2F_1}\...
- 12.4k
10
votes
1
answer
632
views
Closed-form of the hypergeometric function ${_4F_3}\left(\begin{array}c1,1,\tfrac54,\tfrac74\\\tfrac32,2,2\end{array}\middle|\,-t\right)$
Inspired by this question and by using Mathematica the following conjecture seems to be true for all nonzero complex $t$ number:
$${_4F_3}\left(\begin{array}c1,1,\tfrac54,\tfrac74\\\tfrac32,2,2\end{...
- 12.4k
9
votes
1
answer
616
views
Prove this closed-form of sum of ${_4F_3}$ hypergeometric functions
I think the following identity is true. How could we prove it?
$${_4F_3}\left(\begin{array}c 1,1,1,1 \\\tfrac54,2,2\end{array}\middle|\,1\right) + 3\,{_4F_3}\left(\begin{array}c\tfrac12,\tfrac12,1,1\\...
- 12.4k
9
votes
2
answers
473
views
Closed-form of the sequence ${_2F_1}\left(\begin{array}c\tfrac12,-n\\\tfrac32\end{array}\middle|\,\frac{1}{2}\right)$
Is there a closed-form of the following sequence?
$$a_n={_2F_1}\left(\begin{array}c\tfrac12,-n\\\tfrac32\end{array}\middle|\,\frac{1}{2}\right),$$
where $_2F_1$ is the hypergeometric function and $n ...
- 12.4k
12
votes
3
answers
1k
views
Prove ${_2F_1}\left({{\tfrac16,\tfrac23}\atop{\tfrac56}}\middle|\,\frac{80}{81}\right)=\frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$
I've found the following hypergeometric function value by numerical observation. The identity matches at least for $100$ digits.
$${_2F_1}\left(\begin{array}c\tfrac16,\tfrac23\\\tfrac56\end{array}\...
- 12.4k