I am interested in the series 
$$\sum\_{n\geq 1}I\_n(x)\lambda^n$$
which is not the full generating series of the modified Bessel function of the first kind because it starts from $n=1$ and not at $-\infty$.
If we can not find a closed form for this series, what relevant information can we extract from it?



What can we know about "the half" of the generating series of Bessel function
The modified Bessel functions satisfy the recurrence


$$ I\_n(x) - \frac{2(1+n)}{x} I\_{n+1}(x) - I\_{n+2}(x) = 0 $$
which translates to a first-order differential equation for 
$g(\lambda) = \sum\_{n=0}^\infty I\_n(x) \lambda^n$ (note that I'm including $n=0$ in this sum):


$$ 2 \lambda^2 g'(\lambda) + (1-\lambda^2) x g(\lambda) = x I\_0(x) + \lambda x I\_1(x) $$


Thus we get $g(\lambda) = \exp(x (\lambda+1/\lambda)/2) u(\lambda)$ where


$$ u'(\lambda) = \frac{x}{2 \lambda^2} \exp(-x(\lambda+1/\lambda)/2) (I\_0(x) + \lambda I\_1(x)) $$


