A pair of quasi-definite moment functionals ${u_0,u_1}$ is a generalized coherent pair if monic orthogonal polynomials ${P_n(x)}_{n=0}^∞$ and ${R_n(x)}_{n=0}^∞$ relative to $u_0$ and $u_1$, respectively, satisfy a relation
◁수식 삽입▷(원문을 참조하세요)
where $σ_n$ and $τ_n$ 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.
Secondly, for a quasi-definite moment functional σ and nonzero polynomials A(x) and D(x), we define another moment functionals τ by the relations
$D(x)τ=σ, τ=A(x)σ.$
In other words, τ is obtained from σ by a linear spectral transform. The necessary and sufficient conditions for τ to be quasi-definite are introduced. When τ is also quasi-definite, we also introduce a simple representation of orthogonal polynomials relative to τ in terms of orthogonal polynomials relative to σ.
Then we define another moment functional τ by the relation
$D(x)τ=A(x)σ.$
In other words, τ is obtained from σ by a linear spectral transform. We find necessary and sufficient conditions for τ to be quasi-definite when D(x) and A(x) have no non-trivial common factor. When τ is also quasi-definite, we also find a simple representation of orthogonal polynomials relative to τ in terms of orthogonal polynomials relative to σ. We also give two illustrating examples when σ is the Laguerre or Jacobi moment functional.