2 Generalized Cauchy - Stieltjes Transforms and Markov Transforms

We express the generalized Cauchy-Stieltjes transforms of some particular Beta distributions depending on a positive parameter λ as λ-powered Cauchy-Stieltjes transforms of some probability measures. The Cauchy-Stieltjes transforms of the latter measures are shown to be the geometric mean of the Cauchy-Stieltjes transform 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 and the arcsine distributions. We first write down its moments through a terminating series 3F2, however they are not positive integer-valued in general thereby no general combinatorial interpretation holds. Then, we compute the inverse of its Cauchy-Stieltjes transform for λ = 2, 3, 4. 1. Motivation Let λ > 0 and μλ a probability measure (possibly depending on λ) with finite all order moments. The generalized Cauchy-Stieltjes transform of μλ is defined by ∫