There are $N$ non-negative random variables, following independently and identically inverse Gamma distribution with shape parameter $\alpha$ and scale parameter $\beta$ (https://en.wikipedia.org/wiki/Inverse-gamma_distribution). Let $X$ be the sum of those $N$ random variables.
What are the cumulative distribution function (CDF) and probability density function (PDF) of $X$ in closed-forms?
I have a try, as follows: The characteristic function (CF) of $X$ is obviously the power $N$ of the CF of a single random variable. However, when I performed the inverse transform of CF of $X$, there is a power $N$ of modified Bessel function of the second kind in the complex integration, which is very hard to derive a closed-form expression. It may be solved by using the multivariate hypergeometric function or more advanced functions, but I am not good at this field.