How can I find any/all function which satisfies $P(x,y)=P(a,y)P(x-a,y+a)$ for all $x>a>0$?
$P(x,y)$ is the probability that a random walk on the real line starting at 0 reaches -x before reaching y. So we also have $P(cx,cy)=P(x,y)$
By dividing it up into infinitesimal intervals I might be able to solve it if I could find the limit of $(c^2/m)^m Pochhammer[1 + m/c^2, m]$ as m goes to infinity, but mathematica coouldnt tackle it.
How do we obtain P(x,y)?