$$\frac{\pi^4}{1440}-\frac{\pi^2}{3}\log^2(2)+\frac{1}{24}\log^4(2)+\frac{7}{24}\pi^2\log^2(2)+2\log(2) \zeta(3)-\frac{7}{4}\log(2)\zeta(3)+\operatorname{Li}_4\left(\frac{1}{2}\right)$$

Find all polynomials with real coefficients that satisfy $$(x^2-6x+8)P(x)=(x^2+2x)P(x-2)\forall x\in\Bbb R$$ My work; $$\frac{P(x)}{P(x-2)}=-\frac{4}{x-2}+\frac{12}{x-4}+1\tag{1}$$ $$\frac{P(x-2)}{P(x)}=-\frac{12}{x+2}+\frac{4}{x}+1\tag{2}$$ I also factorised the two known polynomial which di...

HINT: Consider $\displaystyle \sum\limits_{n=1}^{\infty} \frac{H_n^{(2)}}{2^nn^2}=\sum\limits_{n=1}^{\infty} \frac{H_{n-1}^{(2)}}{2^nn^2}+\operatorname{Li}_4\left(\frac{1}{2}\right)$ and then express the remaining sum as a double integral. After some work, you get $$\int_0^1 \frac{\displaystyle...

I remember that some time ago I was asking this question Evaluate $\int_0^1\ln(1-x)\ln x\ln(1+x) \mathrm{dx}$ , and now, while I was making a review, I asked myself if we can get the closed form of $$\int_0^1(\ln(1-x)\ln(1+x)\ln(x))^2\,dx$$ by using the similar tools as in that proof. The probl...

Here is a challenging double series question I wanna share with you $$\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \frac{H_n}{kn (k+n)^3}$$ together with the question: what tools would you like employ for computing it?

$$F(\theta)=\sin \theta \int_{-l}^{l}e^{-ikz\cos \theta}f(z)\,dz$$ is it possible to approximate both $f(z)$ and $F(\theta)$ with Fourier Series or Taylor series, then find its coefficient and check out whether it is a good approximation or not? We don't know what the $f(z)$ is, but we know $|z|\...