I'm doing my best to understand the proof of Skolem-Mahler-Lech theorem from https://static1.squarespace.com/static/57bf2a6de3df281593b7f57d/t/57bf600d4402430af257845a/1472159758066/kiddiezeroes.pdf. However I'm not really sure how the $P_k(n)$ polynomials are defined (they appear in the middle of last page).
I've tried to calculate directly $P_1, P_2, P_3, ...$ with no effort. Could somebody give me a hint how to proceed? That's the last thing stopping me from proving the result.