Mock theta functions and characters of N=3 superconformal modules IV

In this paper we obtain explicit formulas for mock theta functions $\Phi^{[m,s]}(\tau, z_1, z_2,t)$ $(m \in \frac12 \mathbf{N}, s \in \frac12 \mathbf{Z})$ by using the coroot lattice of the Lie superalgebra $D(2,1,a)$ and the Kac-Peterson's identity. As its application, we study the branching functions of tensor products of N=3 modules and prove the formula conjectured in the previous paper.