Fourier-Bessel functions and the many asymptotics of orthogonal polynomials of singular continuous measures

We study the Fourier transform of polynomials in an orthogonal
family, taken with respect to the orthogonality measure. Mastering the asymptotic properties of these transforms, that we call Fourier--Bessel functions, in the argument, the order, and in
certain combinations of the two is required to solve a number of
problems arising in quantum mechanics. We present known results, new approaches and open conjectures, hoping to justify our belief that the importance of these investigations extends beyond the application just mentioned, and may involve interesting discoveries.