User:Vladimir Reshetnikov/Sandbox

Let $B\left(z;\ a,b\right)$ denote the incomplete beta function: $$B\left(z;\ a,b\right)=\int_0^zu^{a-1}(1-u)^{b-1}du.$$ It can be expressed in terms of the hypergeometric function: $$B\left(z;\ a,b\right)=\frac{z^a}a{_2F_1}(a,1-b;\ a+1;\ z)$$ and the corresponding integral: $$B\left(z;\ a,b\right)=\frac{z^a\,\Gamma(a)}{\Gamma(a+b)\,\Gamma(1-b)}\int_0^1\frac{(1-x)^{a+b-1}}{(1-x\,z)^a\,x^b}dx.$$

Conjecture 1
Let $\alpha$ be the unique real algebraic number satisfying $$\begin{cases}\alpha^4-540\,\alpha^3+270\,\alpha^2-972\,\alpha+729=0\\\alpha<1\end{cases}$$ It can be expressed in radicals $$\alpha=\left(\sqrt6\,\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2$$ Then, conjecturally, $$B\left(\alpha;\ \frac12,\frac13\right)\stackrel?=\frac{\sqrt\pi}2\cdot\frac{\Gamma\left(\frac13\right)}{\Gamma\left(\frac56\right)}$$

Conjecture 2
Let $\alpha$ be the unique real algebraic number satisfying $$\begin{cases}\alpha^3-99\,\alpha^2+243\,\alpha-81=0\\\alpha<1\end{cases}$$ It can be expressed in radicals $$\alpha=33-\frac{42\,\left(1+\sqrt{-3}\right)\,\sqrt[3]2}{\sqrt[3]{37+\sqrt{-3}}}-3\,\left(1-\sqrt{-3}\right)\,\sqrt[3]4\,\sqrt[3]{37+\sqrt{-3}}$$ Then, conjecturally, $$B\left(\alpha;\ \frac12,\frac13\right)\stackrel?=\sqrt\pi\cdot\frac{\Gamma\left(\frac43\right)}{\Gamma\left(\frac56\right)}$$

Conjecture 3
Let $\alpha$ be the unique real algebraic number satisfying $$\begin{cases}\alpha^9-6129\,\alpha^8-63180\,\alpha^7-1698084\,\alpha^6+5690574\,\alpha^5-21611934\,\alpha^4+28579716\,\alpha^3-10628820\,\alpha^2+4782969\,\alpha-4782969=0\\\alpha<1\end{cases}$$ Then, conjecturally, $$B\left(\alpha;\ \frac12,\frac13\right)\stackrel?=2\,\sqrt\pi\cdot\frac{\Gamma\left(\frac43\right)}{\Gamma\left(\frac56\right)}$$

Conjecture 4
Let $\alpha$ be the unique algebraic number satisfying $$\begin{cases}\alpha^8+1224\,\alpha^7-67284\,\alpha^6+328536\,\alpha^5-1115370\,\alpha^4+367416\,\alpha^3+1338444\,\alpha^2-1417176\,\alpha+531441=0\\0<\alpha<1\end{cases}$$ It can be expressed in radicals $$\alpha=\frac1{7810}\left(-1194930+749760\sqrt3+140580\sqrt{158-91\sqrt3}\\-12\sqrt{3905\left(5494335-3163050\sqrt3-650436\sqrt{158-91\sqrt3}+35145\left(158-91\sqrt3\right)+342\left(158-91\sqrt3\right)^{3/2}+403872\sqrt{3\left(158-91\sqrt3\right)}\right)}\right)$$ Then, conjecturally, $$B\left(\alpha;\ \frac12,\frac13\right)\stackrel?=\frac{3\,\sqrt\pi}4\cdot\frac{\Gamma\left(\frac13\right)}{\Gamma\left(\frac56\right)}$$

Conjecture 5
Let $\alpha$ be the unique real algebraic number satisfying $$\begin{cases}\alpha^8-2088\,\alpha^7+64908\,\alpha^6+21384\,\alpha^5+1917270\,\alpha^4-5616216\,\alpha^3+7007148\,\alpha^2-4251528\,\alpha+531441=0\\\alpha<1\end{cases}$$ It can be expressed in radicals $$\alpha=\frac1{115522}\left(\left(835472448-433427202\,\sqrt5\right)\,\gamma-\sqrt\beta-63388\,\gamma^3+1039698\,\eta\right),$$ where $$\beta=\left(63388\,\gamma^3+18\,\gamma\,\left(24079289\,\sqrt5-46415136\right)-1039698\,\eta\right)^2+6238188\,\left(2842\,\gamma^3+2\,\gamma\,\left(9712963\,\sqrt5-18736651\right)-57761\,\eta\right),$$ $$\gamma=\sqrt{3}\,\sqrt{4745-2122\,\sqrt5},$$ $$\eta=29+13\,\sqrt5.$$ Then, conjecturally, $$B\left(\alpha;\ \frac12,\frac13\right)\stackrel?=\frac{\sqrt\pi}5\cdot\frac{\Gamma\left(\frac13\right)}{\Gamma\left(\frac56\right)}$$

Conjecture 6
Let $\alpha$ be the unique real algebraic number satisfying $$\begin{cases}5\,\alpha^4+360\,\alpha^3-1350\,\alpha^2+729=0\\0<\alpha<1\end{cases}$$ It can be expressed in radicals $$\alpha=3\left(-6+3\sqrt5-\sqrt{6\left(13-\frac{29}{\sqrt5}\right)}\right)$$ Then, conjecturally, $$B\left(\alpha;\ \frac12,\frac13\right)\stackrel?=\frac{3\,\sqrt\pi}5\cdot\frac{\Gamma\left(\frac13\right)}{\Gamma\left(\frac56\right)}$$