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We express generalized Cauchy-Stieltjes transforms of some particular Betadistributions (of ultraspherical type generating functions for orthogonalpolynomials) as a powered Cauchy-Stieltjes transform of some measure. Forsuitable values of the power parameter, the latter measure turns out to be aprobability measure and its density is written down using Markov transforms.The discarded values give a negative answer to a deformed free probabilityunless a restriction on the power parameter is made. A particular symmetricdistribution interpolating between Wigner and arcsine distributions isobtained. Its moments are expressed through a terminating hypergeometric seriesinterpolating between Catalan and shifed Catalan numbers. for small values ofthe power parameter, the free cumulants are computed. Interesting opne problemsrelated to a deformed representation theory of the infinite symmetric group andto a deformed Bozejko's convolution are discussed.

On generalized Cauchy-Stieltjes transforms of some Beta distributions