Generalized coherent pairs

A pair of quasi-definite moment functionals {u(0),u(1)} is a generalized co herent 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, satis fy 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 semi classical. 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 func tionals. Also some illustrating examples and two ways of generating general ized coherent pairs are given. We also discuss the corresponding Sobolev or thogonal polynomials, (C) 2001 Academic Press.