Quadrature formulas for integrals transforms generated by orthogonal polynomials

By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, their asymptotic expressions and bilinear generating functions, we obtain quadrature formulas for the integral transforms generated by the classical orthogonal polynomials. These integral transforms, related to the so-called Poisson integrals, correspond to a fractional Fourier transform in the case of Hermite polynomials, a Bessel transform in the case of Laguerre polynomials and an Appell transform in the case of Jacobi polynomials. We also give a nonclassical example in which a quadrature formula for an integral transform generated by Bateman–Pasternak polynomials is obtained. The kernel of this transform is related to a product of Gauss hypergeometric functions.