What can we know about "the half" of the generating series of Bessel function

Question

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?

Answer 1

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

$$ u'(\lambda) = \frac{x}{2 \lambda^2} \exp(-x(\lambda+1/\lambda)/2) (I_0(x) + \lambda I_1(x)) $$