Polynomial solutions of differential–difference equations Original Research Article

We investigate the zeros of polynomial solutions to the differential–difference equation View the MathML sourcePn+1=AnPn′+BnPn,n=0,1,… Turn MathJax on where AnAn and BnBn 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 including sequences of classical orthogonal polynomials as well as Euler–Frobenius, Bell and other polynomials.