Any solutions to the functional equation $f(x) = f(x-1)-f(x+1)-ix^3f(x)$?

Question

Are there any solutions to this functional equation:

$$f(x) = f(x-1)-f(x+1)-ix^3f(x)$$

I am not familiar with functional equations and have no idea where to start. All I can say is that $f(0) = f(-1)-f(1)$.

Wolfram alpha gives a recurrence relation for $f(n), n\in \mathbb{N}$ provided we know $f(0),f(1)$.

Any suggestions?

Answer 1

Hint:

Take inverse Mellin transform:

$\mathcal{M}^{-1}_{x\to s}\left\{f(x)\right\}=\mathcal{M}^{-1}_{x\to s}\left\{f(x-1)\right\}-\mathcal{M}^{-1}_{x\to s}\left\{f(x+1)\right\}-\mathcal{M}^{-1}_{x\to s}\left\{ix^3f(x)\right\}$

$F(s)=\left(\dfrac{1}{s}-s\right)F(s)+i\left(s\dfrac{d}{ds}\right)^3F(s)$

Which converts to an third-order ODE