Recently, I was working on a certain recursive formula in 2 variables which I was able to simplify to the below $$F(x,y;p)=1+pF(x-1,y;p)+(1-p)F(x+1,y;p)$$ with $p$ being some real constant parameter with $p\in(0,1)$.
However, I have been having trouble solving for the definition of $F$ from here. I have the boundary condition that $F(x,y;p)=0$ for $x=\pm y$. I also know that $F(x,y;\frac{1}{2})=y^2-x^2$.
Given that I have not been able to proceed beyond this, is there a general solution for $F(x,y;p)$ for all $p\in(0,1)$?