Suppose one wants to solve a recurrence relation of the form
$$ c(m+1) - c(m)/f(m) = -g(m) $$ for $c(m)$. The general solution can be given by $$ c(m) = \left(\prod_{\mu=0}^{m-1}\frac{1}{f(\mu)}\right)\left(c_0 - \sum_{k=0}^{m-1}g(k)\prod_{\nu=0}^{k}f(\nu)\right) $$
However, this doesn't, at least in this form, immediately work when I try to extend the solution to half-integer arguments. I also need to modify it to work with negative arguments. Is there a general process that one can follow to analytically continue the solutions to a recurrence to half-integer arguments? Perhaps rephrase the original recurrence? Or is there no good way to do this?
I would also accept willingly as an answer conditions for when it is or is not possible to do this, or even a vague outline of such conditions.