I know that for infinite series and $|z|<1$ there exists a confluent hypergeometric expression

$ \sum_{k=0}^{\infty} \frac{z^k}{k!k!} = F_{1}[;1;z] $

This is not very helpful though, and I 'd like to know if it is possible to get some asymptotic expansion for this function and if there exists some general approach to bounding hypergeometric functions asymptotically.