Partitions with Parts in a Finite Set and with Parts Outside a Finite Set

In a recent paper, M. B. Nathanson [Nathanson] finds the highest order term of the asymptotic formula for the number of partitions with parts in a finite set. This is done without referring to the partial fraction expansion of the generating function. In this paper, we essentially find the entire asymptotic formula without using partial fractions. This is done by the “metaphysical method” described in [Almkvist 98]. There the asymptotic formula was found using Fourier transformations of distributions in a rather obscure way. But when the generating function is rational, it is shown here that the method is perfectly sound. This is based on the following result: