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Densities of the Raney distributions
We prove that if and then the sequence
, , is positive definite, more
precisely, is the moment sequence of a probability measure with
compact support contained in . This family of measures encompasses
the multiplicative free powers of the Marchenko-Pastur distribution as well as
the Wigner's semicircle distribution centered at . We show that if
is a rational number, , then is absolutely continuous and
its density can be expressed in terms of the Meijer and the
generalized hypergeometric functions. In some cases, including the
multiplicative free square and the multiplicative free square root of the
Marchenko-Pastur measure, turns out to be an elementary function
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