Quadratures and integral transforms arising from generating functions

Abstract

By using the explicit form of the eigenvectors of the finite Jacobi matrix associated to a family of orthogonal polynomials and some asymptotic expressions, we obtain quadrature formulas for the integral transforms arising from linear generating functions of the classical orthogonal polynomials. As a bypass product, we obtain simple and accurate Riemann–Steklov quadrature formulas and as an application of this quadrature formalism, we obtain the relationship between the fractional Fourier transform and the canonical coherent states.