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)(infinity) and {R-n(x)}(n=0)(infinity) relative to u(0) and u(1), respectively, satisfy a relation R-n(x) = (1)/(n+1)DeltaP(n+1)(x) - (sigman)/(n)DeltaP(n)(x) - (taun-1)/(n-1)DeltaP(n-1)(x), n greater than or equal to 2, where sigma(n) and tau(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.