  For any variable number, a non-stationary Ruijsenaars function was recently
introduced as a natural generalization of an explicitly known asymptotically
free solution of the trigonometric Ruijsenaars model, and it was conjectured
that this non-stationary Ruijsenaars function provides an explicit solution of
the elliptic Ruijsenaars model. We present alternative series representations
of the non-stationary Ruijsenaars functions, and we prove that these series
converge. We also introduce novel difference operators called ${\mathcal T}$
which, as we prove in the trigonometric limit and conjecture in the general
case, act diagonally on the non-stationary Ruijsenaars functions.
