On the relation between Darboux transformations and polynomial mappings

Abstract Let d μ be a probability measure on [ 0 , + ∞ ) such that its moments are finite. Then the Cauchy–Stieltjes transform S of d μ 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 ) ↦ λ S ( λ 2 ) , 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.

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