Occasion: Workshop on Computer Algebra in Combinatorics
Place: Erwin Schroedinger Institute, Vienna
Certain combinatorial sequences can be encoded as diagonals of multivariate rational functions. In this talk, we will highlight some properties of the coefficients of such rational functions. In particular, we report on recent joint work with Frits Beukers and Marc Houben, in which we investigate necessary as well as sufficient conditions under which all of the coefficients satisfy Gauss congruences. We also connect with the question of positivity of rational functions, which goes back to Szegö as well as Askey and Gasper. Inspired by a result of Gillis, Reznick and Zeilberger, we investigate the relation between positivity of a rational function and the positivity of its diagonal, and suggest the possibility of an algorithmic approach to positivity of a certain class of functions. This part of the talk is based on joint work with Wadim Zudilin.
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