TITLE: Asymptotic bounds for a confluent hypergeometric function $F_{1}[;1;x]$
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: Although not directly for $z < 1$, here is a general paper that might help with some of the techniques that one can use:
The confluent hypergeometric functions $M(a; b, z)$ and $U(a; b, z)$ for large $b$ and $z$ by J. L. López and P. J. Pagola.
Also, if not this paper, please take a look at other papers by López; from what I heard, he is an expert on asymptotics of hypergeometric functions.