On the completeness and minimality of sets of Bessel functions in weighted $L^2$-spaces

We establish a criterion for the completeness and minimality of the system $(x^{-p-1}\sqrt{x\rho_k}J_\nu(x\rho_k):k\in\mathbb{N})$ in the space $L^2((0;1); x^{2p}dx)$ where $J_\nu$ is the Bessel function of the first kind of index $\nu\geqslant1/2$, $p\in\mathbb{R}$ and $(\rho_k : k\in\mathbb{N})$ is a sequence of distinct nonzero complex numbers.