I am looking for a closed-form, analytic solution to the following recursive formula:
$g(m) = -\sigma + \frac{\sigma \kappa}{1 - \beta}\sum_{l=1}^{m}(1 - {\beta}^{l})g(m-l), \forall m>0$
where $g(0) = -\sigma$ - i.e. I am trying to find a general expression for $g(m)$. I tried to "find a pattern" by deriving g(1), g(2), etc. but I wasn't able to find one. I'm sure there is a method to solving equations such as this one, does anyone have any idea how to do it?
Let $G(z) = \sum_{m=0}^\infty g(m) z^m$, and using your formula you should be able to write $G(z)$ as a rational function of $z$ with quadratic denominator. The "closed form" expression for the Maclaurin series of this will be a bit complicated, involving the roots of that denominator.