  We prove that if $p\ge 1$ and $0< r\le p$ then the sequence
$\binom{mp+r}{m}\frac{r}{mp+r}$, $m=0,1,2,...$, is positive definite, more
precisely, is the moment sequence of a probability measure $\mu(p,r)$ with
compact support contained in $[0,+\infty)$. This family of measures encompasses
the multiplicative free powers of the Marchenko-Pastur distribution as well as
the Wigner's semicircle distribution centered at $x=2$. We show that if $p>1$
is a rational number, $0<r\le p$, then $\mu(p,r)$ is absolutely continuous and
its density $W_{p,r}(x)$ 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, $W_{p,r}(x)$ turns out to be an elementary function.
