  When the distribution of the inter-arrival times of a renewal process is a
mixture of geometric laws, we prove that the renewal function of the process is
given by the moments of a probability measure which is explicitly related to
the mixture distribution. We also present an analogous result in the continuous
case when the inter-arrival law is a mixture of exponential laws. We then
observe that the above discrete class of renewal processes provides a solvable
family of random polymers. Namely, we obtain an exact representation of the
partition function of polymers pinned at sites of the aforementioned renewal
processes. In the particular case where the mixture measure is a generalized
Arcsine law, the computations can be explicitly handled.
