Let $\sigma_1, \sigma_2, \dots, $ be the sequence of cumulants of a probability density function $p(x)$. How can we reconstruct $p(x)$ from its cumulants?
P.S. If it helps, you can assume that $p(x)$ is of bounded support in a finite interval $[a,b]$. Also, it is implicit that all the cumulants of $p(x)$ are finite.
The cumulants can be obtained from the cumulant generating function (cgf), which is the logarithm of the moment generating function. The cgf can be used to approximate the density function via the saddlepoint approximation, see How does saddlepoint approximation work?. Other ideas can be found at Constructing a continuous distribution to match $m$ moments and the scholarpedia.
Some examples:
Although the closed-form expression for the PDF cannot be found from cumulants, approximation of the quantile function (also known as the inverse CDF) can be numerically evaluated using the Cornish–Fisher expansion. The CDF values can then be obtained by transposing the quantile function and differentiating them numerically to obtain the PDF see "Finding the PDF given the CDF".
The cumulant generating function is the log of the moment generating function. It is possible then to characterize the distribution with the cumulants.
This in turn can help you obtain the density function.
