On Jacobi functions and multiplication theorems for integrals of Bessel functions

Abstract

In this paper, the product of a Jacobi polynomial and function is shown to generate the Jacobi polynomial. This type of expansion was previously known for all the classical orthogonal polynomials except the Jacobi. The result is then used to obtain generalizations to Watson's [10] multiplication theorem involving integrals of Bessel functions. The integrals considered are of the form ∝ ∞ 0 t 1− λ J v ( tx 1 ) J μ ( tx 2 ) J σ ( tx 3 ) J τ ( tx 4 ) dt .