One-Sided Cauchy–Stieltjes Kernel Families

Characterization of quadratic Cauchy–Stieltjes kernel families based on the orthogonality of polynomials

“…CSK families and associated polynomials. The notations are the ones used in [2]. Suppose ν is a non-degenerate probability measure with support bounded from above.…”

“…Up to affine transformations and powers of free convolution, this class consists of the free Meixner distributions. In [2], Bryc and Hassairi continue the study of CSK families by extending the results to allow measures ν with unbounded support, providing the method to determine the domain of means, introducing the "pseudo-variance" function that has no direct probabilistic interpretation but has similar properties to the variance function. They have also introduced the notion of reciprocity between tow CSK families by defining a relation between the R-transforms of the corresponding generating probability measure.…”

Cauchy–Stieltjes families with polynomial variance functions and generalized orthogonality

“…The Cauchy-Stieltjes Kernel (CSK) families of probability measures were introduced in [11] and extended to non-compact setting in [13]. The constructive approach adopted in these papers is based on an idea of kernel family from an unpublished manuscript [46].…”

“…CSK families with quadratic variance functions were determined in [11], [14], see also [18]. Cubic (pseudo) variance functions with V(0) = 0 have been studied in [13] and they correspond to measures without first moment. In contrast to exponential families, CSK families are not invariant under translation, so cubic variance functions with V(0) = 0 cannot be reduced to the case studied in [13] and require separate investigation.…”

“…We now recall some formulas and assumptions that we will rely upon. It is known (see [13], Proposition 3.1, or [14], (3.4)) that for m = 0…”