  A theorem of Meinardus provides asymptotics of the number of weighted
partitions under certain assumptions on associated ordinary and Dirichlet
generating functions. The ordinary generating functions are closely related to
Euler's generating function $\prod_{k=1}^\infty S(z^k)$ for partitions, where
$S(z)=(1-z)^{-1}$. By applying a method due to Khintchine, we extend Meinardus'
theorem to find the asymptotics of the coefficients of generating functions of
the form $\prod_{k=1}^\infty S(a_kz^k)^{b_k}$ for sequences $a_k$, $b_k$ and
general $S(z)$. We also reformulate the hypotheses of the theorem in terms of
generating functions. This allows us to prove rigorously the asymptotics of
Gentile statistics and to study the asymptotics of combinatorial objects with
distinct components.
