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Seminar on Complex Analysis (Gonchar Seminar)
January 21, 2013 18:00, Moscow, Steklov Mathematical Institute, Room 411 (8 Gubkina)
 


The third-order differential equation for Hermite–Padé polynomials

S. P. Suetin

Steklov Mathematical Institute of the Russian Academy of Sciences

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Abstract: We shall prove the following result.
Theorem. Let function $f(z)=((z-1)/(z+1))^{\alpha}$ 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
$$ (Q_{n,0}+Q_{n,1}f+Q_{n,2}f^2)(z) =O(\frac1{z^{2n+2}}),\quad z\to\infty. $$
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
&-[3(n-1)(n+2)z^2+12\alpha z-(3n(n+1)+8\alpha^2-10)]w'\notag
&+2[n(n^2-1)z+\alpha(3n(n+1)-8)]w=0. \notag \end{align}


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