I am trying to calculate:
$\begin{equation} \sum_{j=0}^{\infty} (\frac{\lambda^j}{j!})(\frac{\lambda^{j+h}}{(j+h)!}) \end{equation}$
And then, I did:
$\begin{equation} \sum_{j=0}^{\infty} (\frac{\lambda^{2j+h}}{(j+h)!(j)!}) = \sum_{j=0}^{\infty} (\frac{\lambda^{2j+h}}{(j+h)!(j)!}) \times (\frac{(j+h+j)!}{(j+h+j)!}) = \sum_{j=0}^{\infty} \begin{pmatrix} 2j+h \\ j \end{pmatrix} \times \frac{\lambda^{2j+h}}{(2j+h)!} \end{equation}$
But how can I prove this result, in fact, converges and to which result?
These are simply modified Bessel functions:
$$I_h(2\lambda)=\sum_{j=0}^\infty\frac{\lambda^j}{j!}\frac{\lambda^{j+h}}{(j+h)!}$$
Absolute convergence trivially follows by comparison to $e^x$'s Maclaurin expansion, the ratio test, etc.