  The paper deals with a three-dimensional family of diffusion processes on an
infinite-dimensional simplex. These processes were constructed by Borodin and
Olshanski (arXiv:0706.1034; arXiv:0902.3395), and they include, as limit
objects, the Ethier-Kurtz's infinitely-many-neutral-allels diffusion model
(1981) and its extension found by Petrov (arXiv:0708.1930).
  Each process X from our family possesses a symmetrising measure M. Our main
result is that the transition function of X has continuous density with respect
to M. This is a generalization of earlier results due to Ethier (1992) and to
Feng, Sun, Wang, and Xu (2011). Our proof substantially uses a special basis in
the algebra of symmetric functions related to Laguerre polynomials.
