The Bessel differential equation has an arbitrary looking form, but a lot is known about it.

$$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $$

Is there a way to derive the Bessel functions from the geometric principles, like the rotational symmetry of the cylinder? A similar situation could be the "quantum harmonic oscillator"

$$ \frac{d^2 \psi}{dx^2} + x^2 \psi = E \psi $$

And we notice the left side factors as $(\frac{d}{dx} + ix)(\frac{d}{dx} - ix)$ + a constant and can use the commuting operators. Even here there no clear "Lie group" action.

Searching shows that Bessel function is related to the affine group of translations in the plane $ \mathbb{E} = SO(2) \ltimes \mathbb{R}^2 $ this is confusing for two reasons:

- Bessel equation is the radial part of Laplace equation $\partial_{xx}^2 + \partial_{yy}^2 + \partial_{zz}^2 f = 0$ in Cylindrial coordinates; so there is not enough symmetry yet
- I found different versions of the raising lowering and operators in one reference
- $J^3 = \partial_y, J^\pm = e^{\pm y} \left( \pm \partial_x - \frac{1}{x} \partial_y \right)$
- $J^3 = z \frac{d}{dz}, J^+ = z, J^- = \frac{1}{z}$

- These references may not be the most modern treatment available, but at least the computations are clear.