Dan Reznik's identities and more
Some conservation laws for families of billiard polygons in ellipses were described in [\textit{D. Reznik} et al., Math. Intell. 42, No. 1, 6--17 (2020; Zbl 1439.37031)]. These were found by means of computer experiments and some of them were proved analytically in [\textit{A. Akopyan} et al., Eur. J. Math. 8, No. 4, 1313--1327 (2022; Zbl 0761.9023]. The authors of this paper offer a different proof using a non-standard generating function for convex billiards in the spirit of the [\textit{M. Bialy}, Nonlinearity 31, No. 5, 2281--2293 (2018; Zbl 1391.37031); \textit{M. Bialy} and \textit{A. Mironov}, Adv. Math. 313, 102--126 (2017; Zbl 1364.37124)]. The approach that the authors present also yields some identities for billiard dynamics inside any smooth convex curve. They also prove other conservation laws found experimentally in [\textit{D. Reznik} et al., loc. cit.].
scientific article; zbMATH DE number 7619034