On the relation between Darboux transformations and polynomial mappings

Let dμdμ be a probability measure on [0,+∞)[0,+∞) such that its moments are finite. Then the Cauchy–Stieltjes transform SS of dμdμ is a Stieltjes function, which admits an expansion into a Stieltjes continued fraction. In the present paper, we consider a matrix interpretation of the unwrapping transformation S(λ)↦λS(λ2)S(λ)↦λS(λ2), which is intimately related to the simplest case of polynomial mappings. More precisely, it is shown that this transformation is essentially a Darboux transformation of the underlying Jacobi matrix. Moreover, in this scheme, the Chihara construction of solutions to the Carlitz problem appears as a shifted Darboux transformation.