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New formulae for higher order derivatives and a new algorithm for numerical integration

This thesis is concerned with the development of new formulae for higher order derivatives, and the algorithmic, numerical, and analytical development of the G (1) n transformation, a method for computing infinite-range integrals. We introduce the Slevinsky-Safouhi formulae I and II with applications, we develop an algorithm for the G (1) n transformation, we derive explicit approximations to incomplete Bessel functions and tail probabilities of five probability distributions from the recursive algorithm for the G (1) n (x) transformation, and we present all extant work on the analysis of the convergence properties of the G (1) n transformation.