  We motivate and prove a series of identities which form a generalization of
the Euler's pentagonal number theorem, and are closely related to specialized
Macdonald's identities for powers of the Dedekind $\eta$--function. More
precisely, we show that what we call ``denominator formula'' for the Virasoro
algebra has ``higher analogue'' for all $c_{s,t}$-minimal models. We obtain one
identity per series which is in agreement with features of conformal field
theory such as {\em fusion} and {\em modular invariance} that require all the
irreducible modules of the series. In particular, in the case of
$c_{2,2k+1}$--minimal models we give a new proof of a family of specialized
Macdonald's identities associated with twisted affine Lie algebras of type
$A^{(2)}_{2k}, k \geq 2$ (i.e., $BC_k$-affine root system) which involve
$(2k^2-k)$-th powers of the Dedekind $\eta$-function. Our paper is in many ways
a continuation of math.QA/0309201.
