Working on my research i came upon the following recurrence relation, which looks extremely similar to the Bessel function recurrence, but has a slight twist where a derivative is present:
Is there a function $D_n(x)$ that satisfies the following:
$$\frac{\partial}{\partial x}\big(D_{n-1}(x)+D_{n+1}(x)\big)=\frac{2n}{\alpha}\cdot D_n(x)$$
and $D_n(x)$ is either even or odd with respect to $x$, where $n\in\mathbb{Z}$ and $\alpha\in\mathbb{R}$.
I think the solution is somehow connected to a sum of Bessel functions.
Example: consider the coefficients of the Laurent series $$\Phi\big(x+(\alpha/2)(z-1/z)\big)=\sum_{n\in\mathbb{Z}}D_n(x)z^n,$$ where $\Phi:\mathbb{C}\to\mathbb{C}$ is an arbitrary (say, entire) function.