All Questions
Tagged with polylogarithm hypergeometric-function
6
questions
25
votes
2
answers
730
views
Definite integral of arcsine over square-root of quadratic
For $a,b\in\mathbb{R}\land0<a\le1\land0\le b$, define $\mathcal{I}{\left(a,b\right)}$ by the integral
$$\mathcal{I}{\left(a,b\right)}:=\int_{0}^{a}\frac{\arcsin{\left(2x-1\right)}\,\mathrm{d}x}{\...
- 30.8k
13
votes
2
answers
522
views
On the integral $\int_{0}^{1/2}\frac{\text{Li}_3(1-z)}{\sqrt{z(1-z)}}\,dz$
This questions is related to my previous one.
I am interested in a explicit evaluation in terms of Euler sums for
$$ \int_{0}^{\pi/4}\text{Li}_3(\cos^2\theta)\,d\theta = \frac{1}{2}\int_{0}^{1/2}\...
- 355k
10
votes
1
answer
478
views
A twisted hypergeometric series $\sum_{n=1}^\infty\frac{H_n}{n}\left(\frac{(2n)!}{4^n(n!)^2}\right)^2$
Question. I was given that $$S=\sum_{n=1}^\infty\frac{H_n}{n}\left(\frac{(2n)!}{4^n(n!)^2}\right)^2=\frac{32}\pi G\ln2+\frac{64}\pi\Im\operatorname{Li}_3\left(\frac{1+i}2\right)-2\ln^22-\frac53\pi^2$$ ...
- 8,679
10
votes
1
answer
578
views
Evaluate $\int_0^1 x^{a-1}(1-x)^{b-1}\operatorname{Li}_3(x) \, dx$
Define
$\small f(a,b)=\frac1{B(a,b)}\int_0^1 x^{a-1}(1-x)^{b-1} \text{Li}_3(x) \, dx$$ $$=\frac a{a+b}{}_5F_4(1,1,1,1,a+1;2,2,2,1+a+b;1)$
Where $a>-1$ and $b>0$.
$1$. By using contour ...
- 8,679
5
votes
1
answer
430
views
Evaluate $\int_0^1 \log (1-x)\ _3F_2\left(1,1,1;\frac{3}{2},\frac{3}{2};x\right) \, dx$
I encountered a hypergeometric integral while investigating harmonic sums
$$\int_0^1 \log (1-x)\ _3F_2\left(1,1,1;\frac{3}{2},\frac{3}{2};x\right) \, dx$$
Based on my experience I suspect a nice ...
- 8,654
20
votes
2
answers
1k
views
Closed form for $\sum_{n=0}^\infty\frac{\Gamma\left(n+\tfrac14\right)}{2^n\,(4n+1)^2\,n!}$
I was experimenting with hypergeometric-like series and discovered the following conjecture (so far confirmed by more than $5000$ decimal digits):
$$\sum_{n=0}^\infty\frac{\Gamma\!\left(n+\tfrac14\...
- 47.3k