Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples
Résumé
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 some 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. Among them figures the symmetric Bernoulli distribution in which case the power of its Stieltjes transform is a generalized Stietljes transform of a probability distribution if and only if the power is greater than one. We close the exposition by considering the generalized Stieltjes transform of a linear functional related with Humbert polynomials and generalizing the symmetric Beta distributions.