$q$-Duality of Prym differentials on compact Riemann surfaces

A general strict $q$-duality of Prym differentials for $q\in\mathbb{Z}$ on compact Riemann surface of genus $g\geqslant1$ and an index of dual complement for strict classical duality (when $q=1$) are introduced. The dimensions of spaces of strictly dual Prym differentials are obtained and their connection with the analytical equations in the Jacobian variety is established.