The third-order differential equation for Hermite–Padé polynomials

We shall prove the following result.
Theorem. Let function $f(z)=\bigl((z-1)/(z+1)\bigr)^{\alpha}$ be holomorphic when $z\notin[-1,1]$, $2\alpha\in\mathbb C\setminus\mathbb Z$ and $f(\infty)=1$. Let $Q_{n,0},Q_{n,1},Q_{n,2}\not\equiv0$ are Hermite–Padé polynomials of degree $n$ for the system $1,f,f^2$, that is
$$ \bigl(Q_{n,0}+Q_{n,1}f+Q_{n,2}f^2\bigr)(z) =O\biggl(\frac1{z^{2n+2}}\biggr),\quad z\to\infty. $$
Then the polinomial $Q_{n,0}$ and functions $Q_{n,1}f$ and $Q_{n,2}f^2$ are the solutions of the following differential equation of third degree:
\begin{align} (z^2-1)^2w''' &+6(z^2-1)(z-\alpha)w''\notag\\ &-\bigl[3(n-1)(n+2)z^2+12\alpha z-(3n(n+1)+8\alpha^2-10)\bigr]w'\notag\\ &+2\bigl[n(n^2-1)z+\alpha(3n(n+1)-8)\bigr]w=0. \notag \end{align}