The Laplace Transform of a Product of Bessel Functions

The Laplace transform of Jμ (at)J_ν (bt)t^λ is an R-function if λ = μ - ν or λ = μ - ν + 1. It is a finite sum of R-functions if μ = ν and λ - 2μ is a nonnegative integer, or if both λ + μ - ν and λ + ν - μ are nonnegative integers. The last result is proved by the techniques of double Dirichlet averages. If 2λ ,2μ, and 2ν are integers, rules are given for determining by-inspection whether the R-functions are rational, algebraic, elementary transcendental, or elliptic. When μ and v are nonnegative integers and λ is an integer, the Laplace transform is a complete elliptic integral of the first or second kind if λ |μ - v| and is conjectured to be a complete elliptic integral of the third kind if λ < |μ - ν |