Generalized valences

We have established that $V(S_p,q;G)$, namely, the collection of all those edges of an arbitrary $n$-vertex hypergraph $G$, whose intersections with set $S_p$, $p$ vertices, has a cardinality $q$, satisfies certain identity relations; in particular, if $v(S_p,q;G)=|V(S_p,q;G)|$, then
$$ v(S_p,q;G)=\sum_{i\ge0}(-1)^iC_{q+1}^q\sum_{S_{q+i}\subset S_p}v(S_{q+i},q+i;G). $$
As applications we derive two new combinatorial identities.