Has this function ever been discussed? Is it an elementary function?

Question

$f^{(n)}(0)=\frac{1}{n!},\ \ n=1,2,\cdots\\$
With Taylor series, this function can also be written as
$f(z)=\sum_{n=0}^{\infty}\frac{x^n}{(n!)^2}$

Has this function ever been discussed? Is it an elementary function? Does it have anything to do with $\Gamma(s)$ (or other famous functions about factorials)?...

Answer 1

$$f(z)=\sum_{n=0}^{\infty}\frac{z^n}{(n!)^2}=I_0\left(2 \sqrt{z}\right)$$ where appears the modified Bessel function of the first kind.