I found on Wikipedia that the first kind of the bessel functions obey the following homogeneous second order difference equation:
$$J_{n+1}(z) = \frac{2n}{z}J_{n}(z) - J_{n-1}(z)$$ Now, I'm trying to find an explicit solution for the non-homogeneous form of this difference equation(k>1): $$c_{k+1} = \frac{2k}{\alpha}c_{k} -c_{k-1} + a \cdot J_{k}(z) $$ With $a,b \in \mathbb{C}$ and $J_{k}(z)$ the bessel function of first kind, evaluated in $z\in \mathbb{C}$
Even though this equation may look daunting, my intuition tells me that it cannot be a coincidence that I have this bessel function as the non-homogeneous part. Since the homogeneous part is the bessel function of first kind(given by the equation above), finding a particular solution is sufficient. But I unfortunately couldn't find any good candidate(if there is one).
Someone who can help?