  We discover new analytic properties of classical partial and false theta
functions and their potential applications to representation theory of
W-algebras and vertex algebras in general. More precisely, motivated by clues
from conformal field theory, first, we are able to determine modular-like
transformation properties of regularized partial and false theta functions.
Then, after suitable identification of regularized partial/false theta
functions with the characters of atypical modules for the singlet vertex
algebra W(2,2p-1), we formulate and prove a Verlinde-type formula for the
fusion rules of irreducible W(2,2p-1)-modules.
