On Polynomial Eigenfunctions of a Hypergeometric-Type Operator

Abstract

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 $x^2-1$.