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 ${\text{Ext}}_{\tau }^1(E,C)$ 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 $\mathrm{\Phi }$ with the zero nilpotent matrix $N_{\mathrm{\Phi }}\mathrm{.}$ For such a \tm module $\mathrm{\Phi }$ we explicitly compute its dual \tm module ${\mathrm{\Phi }}^{\vee }$ as well as its double dual ${\mathrm{\Phi }}^{\vee \vee }\mathrm{.}$ This computation is done in a a subtle way by combination of the \tm reduction algorithm developed by F. Głoch, D.E. K{\k e}dzierski, P. Kraso{ń} [ 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{ń} [On ${\text{Ext}}^1$ 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 $N_{\mathrm{\Phi }}$ is non-zero.