  We prove several infinite families of $q$-series identities for false theta
functions and related series. These identities are motivated by considerations
of characters of modules of vertex operator superalgebras and of quantum
dilogarithms. We also obtain closely related modular identities of the
G\"ollnitz-Gordon-Andrews type. As a byproduct of our identities, we establish
several identities for the Rogers dilogarithm function coming from multi
$q$-hypergeometric series with "double poles".
