  We present a perturbative construction of two kinds of eigenfunctions of the
commuting family of difference operators defining the elliptic Ruijsenaars
system. The first kind corresponds to elliptic deformations of the Macdonald
polynomials, and the second kind generalizes asymptotically free eigenfunctions
previously constructed in the trigonometric case. We obtain these
eigenfunctions as infinite series which, as we show, converge in suitable
domains of the variables and parameters. Our results imply that, for the domain
where the elliptic Ruijsenaars operators define a relativistic quantum
mechanical system, the elliptic deformations of the Macdonald polynomials
provide a family of orthogonal functions with respect to the pertinent scalar
product.
