Solution of Certain Integral Equations with Difference Kernels

We present a method for solving the linear integral equation \[ \int _0^T {R(\exp ( -| x - y| )f(y)dy = S(\exp ( - | t - x| )),\quad 0 \leqq x \leqq T} \] when R and S are given by power series about the origin. The method has a simple derivation and is suitable for numerical work. For T infinite the answer is in an explicit continued product form. A numerical example is discussed.