Moments of inverse of a non-central chi distributed variable

Question

I have a non-central chi variable $r$ with the distribution, \begin{align} p(r) = \frac{r^3\lambda}{(\lambda r)^{3/2}}\exp\left[-0.5(r^2 + \lambda^2)\right]I_{1/2}(r\lambda) \end{align}

I'm looking for the following expectations. \begin{align} E\left[\frac{1}{r}\right]; \quad E\left[\frac{1}{r^2} \right] \end{align}

In other words, is there a closed-form expressions for the integrals of the form, \begin{align} I(m) = \int_{0}^{\infty} \frac{1}{r^m} \frac{r^3\lambda}{(\lambda r)^{3/2}}\exp\left[-0.5(r^2 + \lambda^2)\right]I_{1/2}(r\lambda). \end{align}

Thanks.

Edit :

$I_v(.)$ is the modified Bessel function of the first kind and order $v$.

Answer 1

The density simplifies to

$$p(r)=\frac{\sqrt{\frac{2}{\pi }} r e^{-(\lambda ^2+r^2)/2} \sinh (\lambda r)}{\lambda }$$

where $\sinh$ is the hyperbolic sine function. Integrating with Mathematica results in

$$E(1/r)=\frac{\text{erf}\left(\frac{\lambda }{\sqrt{2}}\right)}{\lambda }$$

$$E(1/r^2)=\frac{\sqrt{2} F\left(\frac{\lambda }{\sqrt{2}}\right)}{\lambda }$$

where $\text{erf}$ is the error function and $F$ is the Dawson F function.

Higher moments don't seem to exist.