Completeness in analytic spaces of subsequences of Laguerre and Jacobi polynomials

In the space $\mathfrak{U}_R$ of all single-valued functions analytic in the circle $|z|<R$ ($0<R\leqslant\infty$), with compact convergence topology, some new tests are found for the completeness of the system of Laguerre polynomials $\{L_{n_j}^{(\alpha)}(z)\}$. An analogous question is considered also in one special analytic space for the Jacobi polynomials.