Probability for asymmetric random walk

Question

How to express this in equation form(in terms of position(x) and time(N)), like the one for symmetric random walk, $\displaystyle P(x,N) = \frac{N!}{(\frac{N+x}{2})! (\frac{N-x}{2})!}p^{\frac{N+x}{2}}q^{\frac{N-x}{2}}$

Also what will be the equation for a random walk with both absorbing and reflecting boundaries on each side(table 2, with absorbing boundary at 3).

Thanks in advance.

Answer 1

Try $\displaystyle2^{-n}\cdot{n\choose \frac{k+n}2}\cdot(1+\mathbf 1_{k\ne0})$, with the convention that $\displaystyle{n\choose i}=0$ when $i$ is not an integer.