  We study higher rank Jacobi partial and false theta functions
(generalizations of the classical partial and false theta functions) associated
to positive definite rational lattices. In particular, we focus our attention
on certain Kostant's partial theta functions coming from ADE root lattices,
which are then linked to representation theory of W-algebras. We derive modular
transformation properties of regularized Kostant's partial and certain higher
rank false theta functions. Modulo conjectures in representation theory, as an
application, we compute regularized quantum dimensions of atypical and typical
modules of "narrow" logarithmic W-algebras associated to rescaled root
lattices. Results in this paper substantially generalize our previous work [19]
pertaining to (1,p)-singlet W-algebras (the sl_2 case).
