  The Kimura equation is a degenerated partial differential equation of
drift-diffusion type used in population genetics. Its solution is required to
satisfy not only the equation but a series of conservation laws formulated as
integral constraints. In this work, we consider a population of two types
evolving without mutation or selection, the so-called neutral evolution. We
obtain explicit solutions in terms of Gegenbauer polynomials. To satisfy the
integral constraints it is necessary to prove new relations satisfied by the
Gegenbauer polynomials. The long-term in time asymptotic is also studied.

Analysis of PDEs;Populations and Evolution