  An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is
concerned with the behavior of the expected value of $\phi(X)$ for large
$\lambda$ where $X$ is a Poisson random variable with mean $\lambda$ and $\phi$
is a function satisfying certain growth conditions. We generalize this by
studying the asymptotics of the expected value of $\phi(X)$ when the
distribution of $X$ belongs to a suitable family indexed by a convolution
parameter. Examples include the problem of inverse moments for distribution
families such as the binomial or the negative binomial.
