New generating function for Birkhoff billiard and applications

I shall discuss a new generating function for Birkhoff billiard (valid also in higher dimensions). It was first found in our joint paper with Andrey E. Mironov. Using this function I will show a new formula for the total length of the polygon corresponding to a periodic orbit. Finally I will specialize to the case of ellipse and give a proof (joint with Sergei Tabachnikov) of remarkable identities obtained recently by Dan Reznik et al (first proved by other method in the paper by Akopyan, Schwartz, Tabachnikov)