Ultimate Positivity of Diagonals of Quasi-rational Functions

The problem to decide whether a given multivariate (quasi-)rational function has only positive coefficients in its power series expansion has a long history. It dates back to Szegö [10], who showed that ((1−Z1)(1−Z2) + (1−Z1)(1−Z3) + (1−Z2)(1−Z3)) for β ≥ 1/2 is positive, in the sense that all its series coefficients are positive, using an involved theory of special functions. In contrast to the simplicity of the statement, the method was surprisingly difficult. This dependency motivated further research for positivity of (quasi-)rational functions. More and more (quasi-)rational functions have been proven to be positive, and some of the proofs are even quite simple [4]. However, there are also others whose positivity are still open conjectures. For instance, the rational function P with