The marcum Q-function is defined by
$$ Q\_m(a,b) = \int^\infty\_b x \left(\frac{x}{a}\right)^{m-1} \exp\left(-\frac{x^2+a^2}{2}\right) 
I\_m\left(a x\right)
\:\mathrm{d} x,$$


where $m\in\mathbb{N}$ , $b\in\mathbb{R}^+$ , $a\in\mathbb{R}^+$ , and $I\_m(.)$ is the $m$-th order modified Bessel function of the first type.


Is it possible to get the derivative of the Q-function with respect to $m$? 



Dertivative of a Special Function with respect to Order
For large $x$, if $a>0$, $I\_m$ behaves asymptotically like $I\_m(ax)\approx e^{ax}/\sqrt{ax}$. Therefore for large $x$ the integrand will look like $x^{m-1/2}e^{-(x-a)^2/2}$. This dies off fast enough that the improper integral converges uniformly in $m$ and you can differentiate inside the integral.


