Ayesha Irfan

Ratios between moments of the characteristic polynomial and its derivative

In a recent paper, Alvarez et al. introduced a ratio between the positive integer moments of the characteristic polynomial and the moments of its derivative evaluated at s=1 over the unitary group. While they obtained several important results concerning this ratio, an explicit formula remained unknown. In this talk, we study a natural generalisation for this ratio in which the moments are considered at general complex arguments. This extension places the original ratio in a broader setting, from which we derive explicit formulae for positive integer moments and uncover connections with certain counting problems such as lozenge tiling of irregular hexagons. We also briefly examine the orthogonal polynomial systems underlying moments of the characteristic polynomial, namely the Heun family, and explain how they offer further insight into the behaviour of these ratios.