Generalized coherent pairs

Abstract

A pair of quasi-definite moment functionals {u(0),u(1)} is a generalized coherent pair if monic orthogonal polynomials {Pn(x)}(n=0)(proportional to), and {R-n(x)}(n=0)(infinity), relative to u(0) and u(1), respectively, satisfy a relation R-n(x) = 1/n+1 P'(n+1)(x)- sigma (n)/n P'(n)(x)-tau (n-1)/n-1 P'(n-1)(X), n greater than or equal to2 where sigma (n) and tau (n) are arbitrary constants, which may be zero. If {u(0),u(1)} is a generalized coherent pair, then u(0) and u(1) must be semiclassical. We find conditions under which either u(0) or u(1) is classical. In such a case, we also determine the types of the "companion" moment functionals. Also some illustrating examples and two ways of generating generalized coherent pairs are given. We also discuss the corresponding Sobolev orthogonal polynomials, (C) 2001 Academic Press.