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The notion of the orbifold Euler characteristic came from physics at the endof 80's. There were defined higher order versions of the orbifold Eulercharacteristic and generalized ("motivic") versions of them. In a previouspaper the authors defined a notion of the Grothendieck ring $K_0^{\rmfGr}(Var)$ of varieties with actions of finite groups on which the orbifoldEuler characteristic and its higher order versions are homomorphisms to thering of integers. Here we define the generalized orbifold Euler characteristicand 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 andgive some analogues of the classical Macdonald equations for the generatingseries of the Euler characteristics of the symmetric products of a space.

Generalized orbifold Euler characteristics on the Grothendieck ring of varieties with actions of finite groups