I would like to put in closed form the integral:
$\int{e^{-k x} I_0(x) dx } $
where $I_\alpha(x)$ is the modified Bessel function of the first kind.
The closest I have found in tables is for k=1
$\int{e^{-x} I_0(x) dx } = x e^{-x} (I_0(x)+I_1(x))$
It would be interesting to see if it is solvable, at least for a numerable set of k values.
The integral arises in a rather fundamental problem: the probability of being inside a ball of defined radious for a bivariate normal. So if it is not solvable, perhaps it worth to define a ad hoc function for it.
Thanks for your interest
Start with $$ \int e^{-k x}\;dx = -\frac{e^{-k x}}{k} $$ by inspection $$ \frac{d^n}{dk^n}\left[-\frac{e^{-k x}}{k}\right]=\sum_{i=1}^{n+1}\frac{(-1)^{n+1}n!}{(n-i+1)!}\frac{x^{n+1-i}e^{-kx}}{k^i} $$ we also have $$ \frac{d^n}{dk^n}\left[e^{-k x}\right]=(-x)^n e^{-kx} $$ generate the operator $$ \hat{O}=\sum_{n=0}^\infty \frac{1}{4^nn!^2}\frac{d^{2n}}{dk^{2n}} $$ apply to both sides of the first equation $$ \hat{O}\left[\int e^{-k x}\;dx\right] = \hat{O}\left[-\frac{e^{-k x}}{k}\right] $$ $$ \int \sum_{n=0}^\infty \frac{1}{4^nn!^2}\frac{d^{2n}}{dk^{2n}}\left[e^{-k x}\right]\;dx = \sum_{n=0}^\infty \frac{1}{4^nn!^2}\frac{d^{2n}}{dk^{2n}}\left[-\frac{e^{-k x}}{k}\right] $$ $$ \int \sum_{n=0}^\infty \frac{1}{4^nn!^2}(-x)^{2n}e^{-kx}\;dx = \sum_{n=0}^\infty \frac{1}{4^nn!^2}\sum_{i=1}^{2n+1}\frac{(-1)^{2n+1}(2n)!}{(2n-i+1)!}\frac{x^{2n+1-i}e^{-kx}}{k^i} $$ we have $$ I_0(x)=\sum_{n=0}^\infty \frac{x^{2n}}{4^nn!^2} $$ $$ \int e^{-kx}I_0(x)\;dx = -e^{-kx}\sum_{n=0}^\infty \frac{(2n)!x^{2n}}{4^nn!^2}\sum_{i=1}^{2n+1}\frac{x^{1-i}}{k^i(2n-i+1)!} $$ $$ \int e^{-kx}I_0(x)\;dx = -e^{-kx}\sum_{n=0}^\infty \frac{(2n)!x^{2n}}{4^nn!^2}\left(-\frac{x}{(1+2n)!}+e^{kx}\frac{\Gamma(2+2n,k x)}{k(kx)^{2n}(1+2n)!} \right) $$ $$ \int e^{-kx}I_0(x)\;dx = xe^{-kx}\,_1F_2\left(\frac{1}{2};1,\frac{3}{2},\frac{x^2}{4}\right) - \sum_{n=0}^\infty \frac{\Gamma(2+2n,k x)}{4^nn!^2 (1+2n)k^{2n+1}} $$ The incomplete gamma function does not help reduce the last sum to a hypergeometric function.