Consider the following more generalized version of the gambler's ruin problem:
A gambler starts with $n$ dollars. On each successive game, the gambler either wins $g$ dollars with probability $p$ or loses $l$ dollars with probability $q = 1-p$. He will stop if he either accumulates $\geq N$ dollars or ends up with $\leq 0$ dollars. What is the probability that he ends up with $\leq 0$ dollars?
I am wondering if there exists a closed-form solution to this. I have tried approaching this problem using the recurrence relation $P[n] = pP[n+g]+qP[n-l]$ with boundary conditions $P[k] = 1 \; \forall k \leq 0$ and $P[k] = 0 \; \forall k \geq N$. However, I was not able to arrive at a solution.
I don't believe so. Zeilberger has some results and computational techniques in https://sites.math.rutgers.edu/~zeilberg/EM20/GamblersRuin.pdf