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\define \cl {\Cal L_{S^1}}
\define \cg {\Cal L_{\Gamma}}
\define \Ga {\Gamma}
\define \la {\lambda}
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\define \bC {\Bbb C}
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\topmatter
\title
A note on  polynomial eigenfunctions of $\frac {d^k}{dx^k}Q_k(x)$
\endtitle

\author G.~M\'asson and B.~Shapiro
\endauthor
\affil
   Department of Mathematics, University of Stockholm, S-10691, Sweden,
{\tt gisli\@matematik.su.se,\; shapiro\@matematik.su.se}\\
\endaffil
\rightheadtext {A note on  polynomial eigenfunctions of $\frac
{d^k}{dx^k}Q_k(x)$}

\abstract

Consider an operator $\frak d_Q=\frac {d^k}{dx^k}(Q_k(x)f(x))$ where Q_k(x)$
$is some fixed polynomial of degree $k$. One can easily see that $\frak d_Q$
has exactly one polynomial eigenfunction $p_n(x)$ in each degree $n\ge 0$
and its eigenvalue $\la_{n,k}$ equals $\frac {(n+k)!}{n!}$. A more
intriguing fact is that all zeros of $p_{n}(x)$ lie in the convex hull of
the set of zeros to $Q_{k}(x)$. In particular, if $Q_{k}(x)$ has only real
zeros then each $p_{n}(x)$ enjoys the same property. We formulate a number
of natural conjectures on different properties of $p_{n}(x)$ based on
computer experiments as, for example, the interlacing property, a formula
for the asymptotic distribution of zeros etc. These polynomial egenfunctions
might be thought as an interesting generalization of the classical
Gegenbauer polynomials with the integer value of the parameter (which
correspond to the case $Q_{2l}(x)=(x^2-1)^l$).

\endabstract

\endtopmatter
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