On Generalized Cauchy-stieltjes Transforms of Some Beta Distributions

We express the generalized Cauchy-Stieltjes transforms (GCST) of some particular Beta distributions depending on a positive parameter λ as λ-powered Cauchy-Stieltjes transforms (CST) of some probability measures. The CST of the latter measures are shown to be the geometric mean of the CST of the Wigner law together with another one. Moreover, they are absolutely continuous and we derive their densities by proving that they are the so-called Markov transforms of compactly-supported probability distributions. Finally, a detailed analysis is performed on one of the symmetric Markov transforms which interpolates between the Wigner (λ = ∞) and the arcsine (λ = 1) distributions. We first write down its moments through a terminating series 3F2 and show that they are polynomials in the variable 1/λ, however they are no longer positive integer-valued as for λ = 1,= ∞ (for instance λ = 2) thereby no general combinatorial interpretation holds. Second, we compute the free cumulants in the case when λ = 2 and explain how to proceed in the cases when λ = 3, 4. Problems of finding a parallel to the representation theory of the infinite symmetric group and an interpolating convolution are discussed. 1. Motivation Let λ > 0 and μλ a probability measure (possibly depending on λ) with finite all order moments. The generalized Cauchy-Stieltjes transform (GCST) of μλ is defined by