In a two-dimensional random walk, a particle can be at any of the points (x, y) which have integer coordinates. The particle starts at (0, 0) and at discrete intervals of time, takes a step of unit size. The steps are independent and equally likely to be any of the four nearest points.

Show that the probability generating function of the time taken to reach the line x + y = m is

(

1 −

p 1 − s2 s

)m for |s| ≤ 1.