### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 035, 13 pages      arXiv:1510.05770      https://doi.org/10.3842/SIGMA.2016.035
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

### Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples

Nizar Demni
IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France

Received December 12, 2015, in final form April 06, 2016; Published online April 12, 2016

Abstract
For two families of beta distributions, we show that the generalized Stieltjes transforms of their elements may be written as elementary functions (powers and fractions) of the Stieltjes transform of the Wigner distribution. In particular, we retrieve the examples given by the author in a previous paper and relating generalized Stieltjes transforms of special beta distributions to powers of (ordinary) Stieltjes ones. We also provide further examples of similar relations which are motivated by the representation theory of symmetric groups. Remarkably, the power of the Stieltjes transform of the symmetric Bernoulli distribution is a generalized Stietljes transform of a probability distribution if and only if the power is greater than one. As to the free Poisson distribution, it corresponds to the product of two independent Beta distributions in $[0,1]$ while another example of Beta distributions in $[-1,1]$ is found and is related with the Shrinkage process. We close the exposition by considering the generalized Stieltjes transform of a linear functional related with Humbert polynomials and generalizing the symmetric Beta distribution.

Key words: generalized Stieltjes transform; Beta distributions; Gauss hypergeometric function; Humbert polynomials.

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