An elementary derivation of the asymptotics of partition functions

Let Sa,b = {an+b:n ≥ 0 } where n is an integer. Let Pa,b(n) denote the number of partitions of n into elements of Sa,b. In particular, we have the generating function, $$ \sum_{n=0}^{\infty} P_{a,b}(n)q^{n}=\prod_{n=0}^{\infty} \frac{1}{(1-q^{an+b})}. $$ We obtain asymptotic results for Pa,b(n) when gcd(a,b) = 1. Our methods depend on the combinatorial properties of generating functions, asymptotic approximations such as Stirling's formula, and an in depth analysis of the number of lattice points inside certain simplicies.