Rational spectral transformations and orthogonal polynomials (Q1372718)

Let \(F(z)= \sum^\infty_{n= 0}{\mu_n\over z^{n+1}}\) be the formal Stieltjes function and let \[ \widetilde F(z)= {A(z) F(z)+ B(z)\over C(z) F(z)+ D(z)}, \] where \(A(z)\), \(B(z)\), \(C(z)\), \(D(z)\) are some polynomials in \(z\), be the rational Stieltjes spectral transform of \(F(z)\). It is shown that the linear \(\widetilde F(z)\) (that is, \(C(z)= 0\)) can be represented as a finite superposition of the Christoffel transform (CT) and the Geronimus-transform (GT), and that the rational Stieltjes transform can be represented as a finite superposition of the following transforms: CT, GT and forward and backward associated transforms.