Closed form of series with factorial-squared denominator?

Solution 1:

Denote

$$f(z)=\sum_{n=0}^\infty\frac{z^n}{(n!)^2}$$

By term-wise differentiation, we find that

$$f'(z)+zf''(z)=f(z)$$

A rather simple differential equation with the general solution

$$f(z)=c_1I_0(2\sqrt z)+c_2K_0(2\sqrt z)$$

where $I_n$ is a modified Bessel function of the first kind, and $K_n$ is a modified Bessel function of the second kind. By using

$$f(0)=f'(0)=1$$

we find that

$$f(z)=I_0(2\sqrt z)$$