I have a process $S_n=X_1+...+X_n$ where all the $X_i$ are iid and $E[X_1]=\mu, Var(X_1)=\sigma^2$ and $\phi(\theta)=Ee^{\theta X_1}$
Now I want to prove two things.
(1) $S_k^2-\sigma^2 k$ for $k\ge0$ is martingale iff $\mu=0$
(2) $e^{\theta S_k}\phi(\theta)^{-k}$ is a martingale for $k\ge 0$
For (1) I want to show that $E[S_{k+1}^2-\sigma^2 (k+1)|F_k]=S_k^2-\sigma^2 k$ iff $\mu=0$. I first want to simplify the LHS: $E[S_{k+1}^2-\sigma^2 (k+1)|F_k]=E[(S_k+X_{k+1})^2|F_k]-\sigma^2 (k+1)$. How can I proceed with the the RHS to include the $\mu$ somewhere?
For (2) I stuck here: $E[e^{\theta S_{k+1}}\phi(\theta)^{-(k+1)}|F_k]=E[e^{\theta S_k}e^{\theta X_{k+1}}E[e^{\theta X_1}]^{-(k+1)}|F_k]$
I assume that the $\sigma$-algebra $F_k=\sigma (X_i,i\le k)$. About $(1)$: You started right: $$E[(S_k+X_{k+1})^2|F_k]=E[S_k^2|F_k]+2S_kE[X_{k+1}|F_k]+E[X^2_{k+1}|F_k]$$. Now since $X_{k+1}$ and $X_k$ are independent, we have $$E[S_k^2|F_k]+2S_kE[X_{k+1}|F_k]+E[X^2_{k+1}|F_k]=S_k^2+2S_k\mu+\sigma^2+\mu^2$$ After all you have now: $$E[(S_k+X_{k+1})^2|F_k]-\sigma^2(k+1)=S_k^2+2S_k\mu-\sigma^2k+\mu$$ Now, $S_k^2+2S_k\mu-\sigma^2k+\mu=S_k^2-k\sigma^2\iff \mu =0$ which proves $(1)$.
Now about the second one: $$E[e^{\theta S_{k+1}}\phi(\theta)^{-(k+1)}|F_k]=\phi(\theta)^{-(k+1)}E[e^{\theta S_{k+1}}|F_k]$$ Therefore all you need to show is $E[e^{\theta S_{k+1}}|F_k]=e^{\theta S_k}\phi(\theta)$
Hint: write $e^{\theta S_{k+1}}=e^{\theta X_{k+1}}e^{\theta S_k}$ and use again independence of $X_{k+1}$ and $F_k$.