Coherent pairs of orthogonal polynomials

Abstract

We will introduce the classical properties of coherent pairs and of symmetrically coherent pairs. That is, for any coherent pair and for any symmetrically coherent pair, in both cases, at least one of the functionals has to be a classical one. And the classifications of coherent pairs with respect to the corresponding orthogonal polynomials and to the corresponding functionals are done with good results and will be introduced in this paper. And then, we will study the generalized coherent pairs that are combined forms of coherent pairs and symmetrically coherent pairs. Let ${P_n}^ ∞_n=0$ and ${R_n}^ ∞_n=0$ be two monic orthogonal polynomial sequences then generalized coherent pair has the form ◁수식 삽입▷(원문을 참조하세요) The problem that we are to solve is whether it can be possible to cover both the classical properties of coherent pairs and that of symmetrically coherent pairs by means of the generalized coherent pairs. And we can see one result of the classical properties that the first functional has to be a classical one in some condition.