Motivic and derived motivic Hirzebruch classes

In this paper we give a formula for the Hirzebruch ${\chi }_y$-genus ${\chi }_y(X)$ and similarly for the motivic Hirzebruch class $T_{y\ast }(X)$ for possibly singular varieties $X$, using the Vandermonde matrix. Motivated by the notion of secondary Euler characteristic and higher Euler characteristic, we consider a similar notion for the motivic Hirzebruch class, which we call a \emph{derived motivic Hirzebruch class}