Narrow Results

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).