Hardy-type theorem for functions orthogonal with respect to their zeros. The Jacobi weight case

Motivated by G. H. Hardy's 1939 results \cite{Hardy} on functions orthogonal with respect to their real zeros $\lambda_{n}, n=1,2,... $, we will consider, within the same general conditions imposed by Hardy, functions satisfying an orthogonality with respect to their zeros with Jacobi weights on the interval $(0,1)$, that is, the functions $f(z)=z^{\nu}F(z), \nu \in \mathbb{R}$, where $F$ is entire and \begin{equation*} \int_{0}^{1}f(\lambda _{n}t)f(\lambda_{m}t)t^{\alpha}(1-t)^{\beta}dt=0,\quad \alpha >-1-2\nu, \beta >-1, \end{equation*}% when $n\neq m$. Considering all possible functions on this class we are lead to the discovery of a new family of generalized Bessel functions including Bessel and Hyperbessel functions as special cases.