  The notion of the orbifold Euler characteristic came from physics at the end
of 80's. There were defined higher order versions of the orbifold Euler
characteristic and generalized ("motivic") versions of them. In a previous
paper the authors defined a notion of the Grothendieck ring $K_0^{\rm
fGr}(Var)$ of varieties with actions of finite groups on which the orbifold
Euler characteristic and its higher order versions are homomorphisms to the
ring of integers. Here we define the generalized orbifold Euler characteristic
and higher order versions of it as ring homomorphisms from $K_0^{\rm fGr}(Var)$
to the Grothendieck ring $K_0(Var)$ of complex quasi-projective varieties and
give some analogues of the classical Macdonald equations for the generating
series of the Euler characteristics of the symmetric products of a space.
