Solution of the Neutral Kimura equation with two integral constraints

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.