We study the rank one Harish-Chandra-Itzykson-Zuber integral in the
limit where \frac{N\beta}{2} \to cNβ2→c, called the high-temperature regime and show that it
can be used to construct a promising one-parameter interpolation family,
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 cumulant-moment relations, a
central limit theorem, a Poisson limit theorem and show several
numerical examples of c-convoluted
distributions.