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This is a re-post of my question from M.SE that remains unanswered for several months. I'm familiar with Kolmogorov–Arnold representation theorem, but AFAIK their construction makes essential use of …

Take some $\nu>0$. Let $J_\nu(x)$ be the Bessel function of the first kind. Let's restrict its domain to $\mathbb R^+$. Is it possible to find a pair of functions $A_\nu(x), \phi_\nu(x):\mathbb R^+\to …

What is known about irrationality of $\pi e$, $\pi^\pi$ and $e^{\pi^2}$?

I asked a question at Math.SE last year and later offered a bounty for it, but it remains unsolved even in the simplest case. So I finally decided to repost this case here: Is it possible to express …

Use formula 16.5.2 from DLMF: $$_3F_2\left(1,1,\frac94;2,2;-1\right)=\int_0^1{_2F_1}\left(1,\frac94;2;-t\right)dt=\frac45\int_0^1\frac{1-(1+t)^{-5/4}}tdt=\\\\\frac{2}{5}\left(8-4\sqrt[4]{8}+\pi-6\ln2+ …

This is a follow-up to my previous question An explicit series representation for the analytic tetration with complex height. Recall the definition $(11)$ from there: $$t(z) = \sum_{n=0}^\infty \sum_ …

Tetration is the next hyperoperation after more familiar addition, multiplication and exponentiation. It can be seen as a repeated exponentiation, similar to how exponentiation can be seen as a repeat …

The tetration is denoted $^n a$, where $a$ is called the base and $n$ is called the height, and is defined for $n\in\mathbb N\cup\{-1,\,0\}$ by the recurrence $$ {^{-1} a} = 0, \quad {^{n+1} a} = a^{\ …

The Fabius function is a smooth monotone function $F:[0,1]\to[0,1]$, satisfying functional equations $$F(0)=0, \quad F(1-x)=1-F(x)\tag1$$ and $$F'(x) = 2 \,F(2 x) \quad \text{for} \,\, 0<x<1/2.\tag2$$ …

Recently I posted a conjecture at Math.SE: $$\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx\stackrel{?}{=}\frac{\pi}{2}(\mu^2-\nu^2),$$ where $J_\mu(x)$ and $Y_\mu(x)$ a …

I asked a question at M.SE a couple of years ago about polylogarithms $\!^{[1]}$$\!^{[2]}$$\!^{[3]}$$\!^{[4]}$ where I conjectured $$720\,\operatorname{Li}_4\!\left(\tfrac12\right)-2160\,\operatorname …