Variant of modified Bessel functions

Question

Consider the integral \begin{align*} g_f(x)=\int_{\phi=0}^{2\pi} f(\phi) ~e^{x cos(\phi)}~\mathrm{d}\phi, \end{align*} where $f(\phi)$ is a probability density functions defined over $[0,2\pi]$, \begin{align*} \forall \phi: f(\phi) \geq 0,~ \int_{\phi=0}^{2\pi} f(\phi)~ \mathrm{d}\phi=1. \end{align*} In case of uniform distribution, i.e. when $\forall \phi: f(\phi) =\frac{1}{2\pi}$, then we know that $g_u(x)=\mathrm{I}_0(x)$, where $\mathrm{I}_0(x)$ is the modified Bessel function of the first kind.

Does there exist other known functions, for other probability density functions $f(\phi)$?

More precisely, in the case of modified Bessel functions of the first kind, we have the following relation:

$$\int_{x=0}^{\infty}x e^{-ax^2}\cdot\mathrm{I}_0(bx)\cdot\mathrm{I}_0(cx)~\mathrm{d}x=\frac{1}{2a}e^{\frac{b^2+c^2}{4a}}~\mathrm{I}_0\left(\frac{bc}{2a}\right).$$

Do we have similar relation (or upper bounds) on

$$\int_{x=0}^{\infty}x e^{-ax^2}\cdot g_{f_1}(bx)\cdot g_{f_2}(cx)~\mathrm{d}x,$$ for other particular choices of $f_1$ and $f_2$? Even for the simpler case where $f_1=f_2$? The ideal answer for me is something proportional to $g_{f_3}(\frac{bc}{2a})$ (or $g_{f_3}(d\frac{bc}{2a})$, for some parameter $d$), for some pdf $f_3$.

P.S. The question was first asked in math.stackexchange. After some times without any response, I deleted it there, and posted it here.

Answer 1

Q: Do there exist other known functions, for other probability density functions $f(\phi)$?

Here are three:

$$f(\phi)=\frac{1}{\pi}\cos^2 \phi\Rightarrow g(x)=\frac{2 [I_1(x)+x I_2(x)]}{x},$$ $$f(\phi)=\frac{4}{\pi}\cos^2 \phi\sin^2\phi\Rightarrow g(x)=\frac{8 (x I_1(x)-3 I_2(x))}{x^2},$$ $$f(\phi)=\frac{1}{2\pi I_0(1)}e^{\sin\phi}\Rightarrow g(x)=\frac{I_0\left(\sqrt{x^2+1}\right)}{I_0(1)}.$$