CAUCHY–STIELTJES FAMILIES WITH POLYNOMIAL VARIANCE
FUNCTIONS AND GENERALIZED ORTHOGONALITY

Włodzimierz Bryc

Raouf Fakhfakh

Wojciech Młotkowski

Abstract: This paper studies variance functions of Cauchy–Stieltjes Kernel (CSK) families
generated by compactly supported centered probability measures. We describe
several operations that allow us to construct additional variance functions from
known ones. We construct a class of examples which exhausts all cubic variance
functions, and provide examples of polynomial variance functions of arbitrary degree.
We also relate CSK families with polynomial variance functions to generalized
orthogonality.

Our main results are stated solely in terms of classical probability; some proofs rely on
analytic machinery of free probability.

2000 AMS Mathematics Subject Classification: Primary: 60E10, 46L54, 62E10;
Secondary: 05A15.

Keywords and phrases: Kernel families, generalized orthogonality, -transform,
-transform, Fuss–Catalan numbers, variance functions, free additive convolution, free
multiplicative convolution.