Let $X_{i}$ be iid discrete random variable with distribution $P(X_{1}=-1)=q$ and $P(X_{1}=i)=f_{i}$ for all $i\geq 0$.
Then consider the random walk $S_{n}=\sum_{i=1}^{n}X_{i}$ and $S_{0}=0$. Let $\lambda_{r}$ be the probability that the probability that the first positive term of the sequence $S_{1},S_2,...$ assumes the value $r$. In otherwords, $\lambda_{r}$ is the probability that the first ladder height takes the value $r$.
Now, problem $3$ asks us to show the following reccurence,
$$\lambda_{r}=f_{r}+q(\lambda_{r+1}+\lambda_{1}\lambda_{r})$$
And problem $4$ asks us in the above setting, if $\gamma_{r}$ denotes the probability that the first non-negative term of $S_1, S_2,. . .$ assumes the value $r$, then we have the following recurrence,
$$\gamma_{r}=f_{r}+\frac{q}{1-\gamma_{0}}\cdot\gamma_{r+1}$$
My attempt:
For the problem $3$, let $A_{r}$ be the event that the first positive term of the sequence $S_{1},S_{2},...$ assumes the value $r$ and I write in the following way:
$\lambda_{r}=P(X_{1}=r)+P(X_{1}=-1,A_{r})$. Now after hitting $-1$, for $A_{r}$ to occur the random walk restarted at $-1$ can be such that the first value strictly above $-1$ it hits is either $r+1$ OR we have that the first value strictly above $-1$ is $1$, in which case, we return to $0$ and then we restart the walk and hit level $r$. The above is written in words so that I don't get lost in notations. Hence by the Strong Markov Property, I can write the above in the form as
$$P(X_{1}=-1,A_{r})= q\lambda_{r+1}+q\lambda_{1}\lambda_{r}$$ which proves the relation for $\lambda_{r}$.
Now for the problem $4$, I can argue that after $X_{1}=-1$, the restarted random walk can either hit the level $r+1$ above $-1$ or it can hit the level $0$ above $-1$ and again we restart at $-1$. So I think we'll get a geometric progression that we after reaching the level $0$ above $-1$, $k$ times, we hit level $r+1$ and this $k$ can be any non negative integer.
So I think we should get the following relation,
$$\gamma_{r}=f_{r}+q(\gamma_{r+1}+\gamma_{0}\gamma_{r+1}+\gamma_{0}^{2}\gamma_{r+1}+...)=f_{r}+\frac{q\gamma_{r+1}}{1-\gamma_{0}}$$
Question: Is my thinking above correct or am I missing some details? If yes, then can someone tell me how to write this up with full rigour as I am struggling a lot to actually come up with the justifications or writing out the reasoning using the law of total probability. To sum up, I think what I did above is correct and I would appreciate it if someone can write out the wordy justification I gave in terms of "measurable" events and give me an expression using the law of total probability.