A recursive algorithm for the G transformation and accurate computation of incomplete Bessel functions

In the present contribution, we develop an efficient algorithm for the recursive computation of the transformation for the approximation of infinite-range integrals. Previous to this contribution, the theoretically powerful transformation was handicapped by the lack of an algorithmic implementation. Our proposed algorithm removes this handicap by introducing a recursive computation of the successive transformations with respect to the order n. This recursion, however, introduces the operator applied to the integrand. Consequently, we employ the Slevinsky–Safouhi formula I for the analytical and numerical developments of these required successive derivatives.Incomplete Bessel functions, which pose as a numerical challenge, are computed to high pre-determined accuracies using the developed algorithm. The numerical results obtained show the high efficiency of the new method, which does not resort to any numerical integration in the computation.