Let $Y1 , Y2,...$ be independent identically distributed random variables with $\mathbb{P}(Y1 =1)=\mathbb{P}(Y1 =-1)=1/2$ and set $Xo=1,Xn =Xo+Y1+...+Yn$ for $n\geq1$. Define; $$H_o= inf\{n\geq0:Xn = O\}$$ $\bullet$ Find the probability generating function $\phi(s) = E(s^{H_0})$.
$\bullet$ Suppose the distribution of $Y1 ,Y2, ...$ is changed to $\mathbb{P}(Y1 = 2) = 1/2, \mathbb{P}(Y1 = -1) = 1/2$. Show that now satisfies $$s\phi^3-2\phi+s= 0$$.
MY ATTEMPT
If we start at 2 then $H_{0}=H_{1}+\tilde{H_{1}}$ where $\tilde{H_{0}}$ is the time taken after 1 to get back to 0. so $$\mathbb{E}_{2}(s^{H_{0}})=\mathbb{E}_{2}(s^{H_1}|H_{1}<\infty)\mathbb{E}_{2}(s^{\tilde{H_{0}}}|H_{1}<\infty)\mathbb{P}_{2}(H_1<\infty)$$
$$=\mathbb{E}_{2}(s^{H_1}\mathbb{1}_{H_{1}<\infty})\mathbb{E}_{2}(s^{\tilde{H_{0}}}|H_{1}<\infty)$$
$$=\mathbb{E}_{2}(s^{H_1})^{2}=\phi(s)^{2}$$
Now we can rewrite the generating function as follows;
$$\phi(s)=0.5\mathbb{E}(s^{H_0}|X_1=2)+0.5\mathbb{E}(s^{H_0}|X_1=0)$$
So we can write $H_{0}=1+\bar{H_{0}}$ where $\bar{H_{0}}$ si time after 1 to get to 0; $$0.5\mathbb{E}(s^{1+H_0}|X_1=2)+0.5\mathbb{E}(s|X_1=0)$$
$$=0.5s \phi^{2}-\phi+0.5s=0$$ and so we get the solution to be $$\phi=\frac{1 +/- \sqrt{1-s^2}}{s}$$
But I cant seem to apply the same logic to the seocnd part fo the question? Where you have jumps that are bigger than one step?
Any help would be appreciated.