  We investigate the zeros of polynomial solutions to the
differential-difference equation \[
P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x), n=0,1,... \] where
$A_{n}$ and $B_{n}$ are polynomials of degree at most 2 and 1 respectively. We
address the question of when the zeros are real and simple and whether the
zeros of polynomials of adjacent degree are interlacing. Our result holds for
general classes of polynomials but includes sequences of classical orthogonal
polynomials as well as Euler-Frobenius, Bell and other polynomials.

Classical Analysis and ODEs