Weil-Barsotti formula for $\mathbf{T}$-modules

In the work of M. A. Papanikolas and N. Ramachandran [A Weil-Barsotti formula for Drinfeld modules, Journal of Number Theory 98, (2003), 407-431] the Weil-Barsotti formula for the function field case concerning $\Ext_{\tau}^1(E,C)$ where $E$ is a Drinfeld module and $C$ is the Carlitz module was proved. We generalize this formula to the case where $E$ is a strictly pure \tm module $\Phi$ with the zero nilpotent matrix $N_\Phi.$ For such a \tm module $\Phi$ we explicitly compute its dual \tm module ${\Phi}^{\vee}$ as well as its double dual ${\Phi}^{{\vee}{\vee}}.$ This computation is done in a a subtle way by combination of the \tm reduction algorithm developed by F. G{\l}och, D.E. K{\k e}dzierski, P. Kraso{\'n} [ Algorithms for determination of \tm module structures on some extension groups , arXiv:2408.08207] and the methods of the work of D.E. K{\k e}dzierski and P. Kraso{\'n} [On $\Ext^1$ for Drinfeld modules, Journal of Number Theory 256 (2024) 97-135]. We also give a counterexample to the Weil-Barsotti formula if the nilpotent matrix $N_{\Phi}$ is non-zero.