Discrete classical orthogonal polynomials

Abstract

We find necessary and sufficient conditions for the difference equation of hypergeometric type L[y] (x) := alpha(x)Delta del y(x)+ beta(x)Delta y(x)=lambda(n)y(x) to have polynomial solutions {p(n)(x)}(n=0)(infinity), which are orthogonal, that is, integral(infinity)(-infinity) P-m(x)P-n(x)d mu(x)= 0, m not equal n, Traditionally, d mu(x) is a positive measure but here we allow it to be a signed measure. We then show that the usual restrictions on parameters in discrete classical orthogonal polynomials can be relaxed. We also derive functional Rodrigues' formula for discrete classical orthogonal polynomials. Finally, we give a nonlinear recurrence relation characterizing discrete classical orthogonal polynomials. This is the first nonlinear characterization of discrete classical orthogonal polynomials whereas several nonlinear characterizations are known for classical orthogonal polynomials.