Short version:
I need to find out what known identity is used to convert the following integral into the formula with Kummer's confluent hypergeometric function, ${}_1F_1(a;b;z)$:
$$ (2\pi\sigma^2)^{-n/2}e^{-\frac{\mu^2}{2\sigma^2}}\int_0^\infty dr\int_0^\pi r A_{n-2}(r\sin\theta)e^{-\frac{r^2}{2\sigma^2}+\frac{\mu r}{\sigma^2}\cos\theta}\frac{\cos\theta}{r} d\theta = \frac{\Gamma\left(\frac{n}{2}+1-\frac{k}{2}\right)}{(2\sigma^2)^{k/2}\Gamma\left(\frac{n}{2}+1\right)}{}_1F_1\left(\frac{k}{2};\frac{n+2}{2};-\frac{|\mu|^2}{2\sigma^2}\right) $$
where $A_{n-1}(r)=\frac{2\pi^{n/2}}{\Gamma(n/2)}r^{n-1}$ is the area of a $(n-1)$-sphere of radius $r$.
Context:
I recently asked in this forum about an expected value of a function of a Gaussian random variable. Someone answered providing a formula that uses ${}_1F_1(a;b;z)$.
The answer also provided the intermediate step of expressing the integral for the expected value in a convenient form that apparently is where the final formula is obtained from. However, I am unable to figure out how the final formula emerges from the integral formula provided by the answer.
I have tried looking into identities of ${}_1F_1(a;b;z)$, and try to better understand the integral, but it seems outside of my mathematical capabilities (I'm a biologist). Thus, I am looking for pointers on to better understand how the two sides of the identity above relate to one another.