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GENERATING FUNCTIONS OF CAUCHY–STIELTJES TYPE FOR ORTHOGONAL POLYNOMIALS

MAREK BOŻEJKO

Instytut Matematyczny, Uniwersytet Wroclawski, Pl., Grunwaldzki 2/4, 50-384 Wroclaw, Poland

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and

NIZAR DEMNI

Faculty of Mathematics, Bielefeld University, Germany

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https://doi.org/10.1142/S0219025709003586Cited by:5 (Source: Crossref)

Abstract

We give a free probabilistic interpretation of the multiplicative renormalization method. As a byproduct, we give a short proof of the Asai–Kubo–Kuo problem on the characterization of the family of measures for which this method applies with h(x) = (1 - x)^-1 which turns out to be the free Meixner family. We also give a representation of the Voiculescu transform of all free Meixner laws (even in the non-freely infinitely divisible case).

Keywords:

AMSC: Primary 46L54, Secondary 60G20

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