<s>[INST] Series involving Marcum Q function I would like to have a better form of this series:
$$\sum_{k=0}^{\infty}\,\frac{1}{k!}\,\left(\frac{ab\sin(c)}{\sqrt{2}}\right)^{2k}\,Q_{k+\frac{3}{2}}\left(ab\cos(c),bx\right)$$
where $Q_m(\alpha,\beta)$ is the generalized Marcum Q-function.
Does anyone an idea on how to proceed, if possible to make it better (e.g. getting rid of the series)?
Thanks!
EDIT:
could the result be just proportional to $Q_{\frac{3}{2}}\left(ab,bx\right)$???
 [/INST] Well, I guess, in some form of. But at the same time you can get a closed form solution.

For simplicity let's set $\alpha=\left(\frac{ab\sin(c)}{\sqrt{2}}\right)^{2}$ and $\beta=ab\cos(c)$
 Taking into account the definition of the generalized Marcum Q-function:
$$
\begin{eqnarray}
\sum_{k=0}^{\infty}\,\frac{\alpha^{k}}{k!}\,\,Q_{k+\frac{3}{2}}\left(\beta,bx\right)&=&
\sum_{k=0}^{\infty}\,\frac{\alpha^{k}}{k!}\,\,\int_{bx}^{\infty} t \left( \frac{t}{\beta}\right)^{k+\frac{1}{2}} \exp \left( -\frac{t^2 + \beta^2}{2} \right) I_{k+\frac{1}{2}} \left( \beta t \right) \ \mathrm dt\\
&=&\frac{\exp \left(\!-\!\frac{\beta^2}{2}\! \right)}{\sqrt{\beta}}\int_{bx}^{\infty} t^{\frac{3}{2}}e^{-\frac{t^2}{2}}\sum_{k=0}^{\infty}\frac{\left(\frac{\alpha t}{\beta}\right)^k}{k!}I_{k+\frac{1}{2}}(\beta t) \ \mathrm dt\\
\end{eqnarray}
$$
Then one can use Gradshteyn and Ryzhik's Table of Integrals, Series, and Products (Second Part):
$$\sum_{k=0}^{\infty}\frac{\xi^k}{k!}I_{k+\nu}(\eta)=\left(\frac{2\xi}{\eta}+1\right)^{-\frac{\nu}{2}}I_{\nu}\left(\sqrt{\eta^2+2\eta\xi}\right), \qquad |2\xi|<|\eta|$$
So
$$
\begin{eqnarray}
\sum_{k=0}^{\infty}\,\frac{\alpha^{k}}{k!}\,\,Q_{k+\frac{3}{2}}\left(\beta,bx\right)
=\frac{\exp \left(\!-\!\frac{\beta^2}{2}\! \right)}{\sqrt{\beta}}\left(1+\frac{2\alpha}{\beta^2}\right)^{-\frac{1}{4}}
\int_{bx}^{\infty} t^{\frac{3}{2}}e^{-\frac{t^2}{2}}I_{\frac{1}{2}}(\sqrt{2\alpha+\beta^2}|t|) \ \mathrm dt=\\
=\text{A} \ \left[\sqrt{\pi } \sqrt{2 \alpha +\beta ^2} e^{\alpha +\frac{1}{2} \left(b^2 x^2+\beta ^2\right)+b x \sqrt{2 \alpha +\beta ^2}}\left(\text{erfc}\left(\frac{ \sqrt{2 \alpha +\beta ^2}+b x}{\sqrt{2}}\right)+\text{erfc} \left(\frac{b x-\sqrt{2 \alpha +\beta ^2}}{\sqrt{2}}\right) \right)+\sqrt{2} \left(e^{2 b x \sqrt{2 \alpha +\beta ^2}}-1\right)\right]
\end{eqnarray}
$$
where $\text{A}=\frac{\exp  \left(-\frac{\beta ^2}{2}-\frac{1}{2} b x \left(2 \sqrt{2 \alpha +\beta ^2}+b x\right)\right)}{2 \sqrt{\pi } \sqrt{\left(2 \alpha +\beta ^2\right) \text{sgn}(\beta)}}$
The last equation was obtained via Mathematica (or you can turn to Gradshteyn and Ryzhik ;)). The equation was simplified assuming that $\alpha>0, \ x>0, \ \beta\in\mathbb{R}$. If not the result gets nastier (with lots of square roots).
I guess you can turn the last integral into some form of the generalized Marcum Q-function, if you'd like to:
$$\int_{bx}^{\infty} t^{\frac{3}{2}}e^{-\frac{t^2}{2}}I_{\frac{1}{2}}(\sqrt{2\alpha+\beta^2}|t|) \ \mathrm dt=(2\alpha+\beta^2)^{\frac{1}{4}}e^{\frac{|2\alpha+\beta^2|}{2}}Q_{\frac{3}{2}}\left(\sqrt{2\alpha+\beta^2},bx\right)$$
So
$$\sum_{k=0}^{\infty}\,\frac{\alpha^{k}}{k!}\,\,Q_{k+\frac{3}{2}}\left(\beta,bx\right)=e^{\alpha}Q_{\frac{3}{2}}\left(\sqrt{2\alpha+\beta^2},bx\right)$$
Noting that $\sqrt{2\alpha+\beta^2}=a \ b $ the result turns out to be even simplier.
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