  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.
