I've encountered the following beast in my research:
$$\frac{f(|x|)}{f(|x-\delta|)}=\operatorname{Exp}\left(a+bx^2-\frac{f(|x+\delta|)}{f(|x|)}\right)$$
Here, $x$ and $\delta\neq0$ are real numbers, and $a$ and $b$ are real constants.
I am wondering whether one can one write down $f(|x|)$ that satisfies the above in terms of functions of $|x|$, $a$, and $b$, removing $\delta$? I'll be happy with the solution involving special functions (hopefully ones that are implemented in Mathematica or MATLAB) -- I've tried using the Lambert W, but got stuck...
Your equation seems to have some problems. Assuming $b\neq 0$, if you set $\delta=0$ you get $\exp\left(a+bx^2\right)$ is a constant which contradicts $b\neq 0$.