Generalized Delta coherent pairs

Abstract

A pair of quasi-definite linear functionals ${u_0,u_1}$ is a generalized $Delta$-coherent pair if monic orthogonal polynomials $${P_n(x)}_{n=0} nfty$$ and $${R_n(x)}_{n=0} nfty$$ relative to $u_0$ and $u_1$, respectively, satisfy a relation $$ R_n(x) = frac{1}{n+1}Delta P_{n+1}(x)-frac{sigma_n}{n}Delta P_n(x)- frac{ au_{n-1}}{n-1}Delta P_{n-1}(x), ~~ ngeq 2,$$ where $sigma_n$ and $ au_n$ are arbitrary constants and $Delta p=p(x+1)-p(x)$ is the difference operator.

We show that if ${u_0,u_1}$ is a generalized $Delta$-coherent pair, then $u_0$ and $u_1$ must be discrete-semiclassical linear functionals. We also find conditions under which either $u_0$ or $u_1$ is discrete-classical.