Generalized Coherent Pairs

A pair of quasi-definite moment functionals u0, u14is a generalized coherent pair if monic orthogonal polynomials PnŽx.4 n 0 and RnŽx.4 n 0relative to u0and u1, respectively, satisfy a relation RnŽx. n 1 1Pn 1Žx. nnPn Žx. n n 11Pn 1Žx., n 2, where n and n are arbitrary constants, which may be zero. If u0, u14is a generalized coherent pair, then u0 and u1must be semiclassical. We find conditions under which either u0or u1is 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