I am trying to express the famous Nuttal Q-function, given as:
$$\mathcal Q_{m,n}(p,q)=\int_q^\infty t^me^{-0,5\left[p^2+t^2\right]}I_n(pt)\;dt$$
where $m$, $n$, $p$, and $q$ are constants and $I_{n}(*)$ is the modified bessel function of the first type, in terms of the Gaussian Q-function, given as:
$$Q(x)=\frac1{\sqrt{2\pi}}\int_x^\infty\exp\left(-\frac{u^2}2\right)\;du$$
So, I carried the following steps, but I believe a mistake occured when I introduced the Gaussian Q-function integral (third line), which will directly affect the rest of the derivations and manipulations.
Can somebody help me with embedding the Gaussian Q-function within the Nuttal Q-function?
$$\begin{align} \mathcal Q_{m,n}(p,q)&=\int_q^\infty t^me^{-0,5\left[p^2+t^2\right]}I_n(pt)\;dt\\ &=e^{-0.5p^2}\int_q^\infty\left\{t^mI_n(pt)\right\}\left\{e^{-0.5t^2}\right\}\;dt\\ &=\sqrt{2\pi}e^{-0.5p^2}\int_p^\infty\left\{\left\{t^mI_n(pt)\right\}\left\{\frac d{dt}\int_t^\infty\frac1{\sqrt{2\pi}}e^{-0.5x^2}\;dx\right\}\right\}\;dt\\ &=\sqrt{2\pi}e^{-0.5p^2}\int_p^\infty\left\{\left\{t^mI_n(pt)\right\}\left\{\frac d{dt}\mathcal Q(t)\right\}\right\}\;dt \end{align}$$