Generalized Cauchy-stieltjes Transforms of Some Beta Distributions

We express 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, these measures are shown to be 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 the are for λ = 1,∞ (for instance λ = 2). 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 deformation of 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 ∫ R 1 (z − x)λ μλ(dx) for nonreal complex z lying in a suitable branch [8, 17, 20]. For λ = 1, it reduces to the (ordinary) Cauchy-Stieltjes transform (CST) which has been of great importance during the two last decades for both probabilists and algebraists due the central role it plays in free probability and representation theories [1, 10]. Moreover, CST were extensively studied and they are handable in the sense that for instance, a complete characterization of those functions is known and one has a relatively easy inversion formula due to Stieltjes [10]. However, their generalized versions are more hard to handle as one may realize from the complicated inversion formulas displayed in [8, 17, 20]. In this paper, we adress the problem of relating 2000 Mathematics Subject Classification. Primary 44A15; 44A35; Secondary 44A60.