Consider the operator
\frakd_Q(f)=\frac {d^k}{dx^k}(Q(x)f(x)),
¶ where
$Q(x)$ is some fixed polynomial of degree $k$. One can easily see
that $\frakd_Q$ has exactly one polynomial eigenfunction $p_n(x)$ in
each degree $n\ge 0$ and its eigenvalue $\la_{n,k}$ equals
$(n+k)!/{n!}$.
A more intriguing fact is that all zeros of $p_{n}(x)$ lie in the
convex hull of the set of zeros to $Q(x)$. In particular, if
$Q(x)$ has only real zeros then each $p_{n}(x)$ enjoys the same
property. We formulate a number of 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 eigenfunctions might be thought of as a
generalization of the classical Gegenbauer polynomials with
half-integer superscript, this case arising when our
$Q(x)$ is an integer power of $x^2-1$.