Mellin Transforms and Asymptotics: Harmonic Sums
Résumé
This survey presents a unified and essentially self-contained approach to the asymptotic analysis of a large class of sums that arise in combinatorial mathematics, discrete probabilistic models, and the average case analysis of algorithms. It relies on the Mellin transform, a close relative of the integral transforms of Laplace and Fourier. The method applies to harmonic sums that are superpositions of rather arbitrary ``harmonics'' of a common base function. Its principle is a precise correspondence between individual terms in the asymptotic expansion of an original function and singularities of the transformed function. The main applications discussed are in the area of digital data structures, probabilistic algorithms, and communication theory.