How to find H(z) and H(k) from a given causal function
It is a Homework problem, so only Hint will be provided. You need to show some work of your own following the hints.

First of all, take $\mathcal Z$-transform of the equation you have. Take care of the delays in time-domain and apply appropriate property of $\mathcal Z$-transform.

Then, try to get the transfer function $H(z)$ from there.

Once you have $H(z)$, figure out a way to convert $H(z)$ into $H(e^{j\omega})$ and then try to proceed for $H[k]$.

Show your work by making updates in your question and we will be clarify the doubts and be able to help you understand.