Multivariate Poincar\'e series for algebras of $S{L}_2$-invariants

Let ${\mathcal{C}}_{\text{mathbi}d},$ ${\mathcal{I}}_{\text{mathbi}d},$ $\text{mathbi}d=(d_1,d_2,\mathrm{.}\mathrm{.}\mathrm{.},d_n)$ be the algebras of join covariants and joint invariants of the $n$ binary forms of degrees $d_1,d_2,\mathrm{.}\mathrm{.}\mathrm{.},d_n\mathrm{.}$ Formulas for computation of the multivariate Poincar'e series $\mathcal{P}({\mathcal{C}}_{\text{mathbi}d},z_1,z_2,\mathrm{.}\mathrm{.}\mathrm{.},z_n,t)$ and $\mathcal{P}({\mathcal{I}}_{\text{mathbi}d},z_1,z_2,\mathrm{.}\mathrm{.}\mathrm{.},z_n)$ are found.