  We study the rank one Harish-Chandra-Itzykson-Zuber integral in the limit
where $\frac{N \beta}{2} \to c $, called the high temperature regime and show
that it can be used to construct a promising one-parameter interpolation, with
parameter $c$ between the classical and the free convolution. This
$c$-convolution has a simple interpretation in terms of another associated
family of distribution indexed by $c$, called the Markov-Krein transform: the
$c$-convolution of two distributions corresponds to the classical convolution
of their Markov-Krein transforms. We derive first cumulants-moments relations,
a central limit theorem, a Poisson limit theorem and shows several numerical
examples of $c$-convoluted distributions.
