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

False Theta Functions and the Verlinde formula