in 2015, marcus, spielman, and srivastava realized that expected characteristic polynomials of sums and products of randomly rotated matrices behave like finite versions of voiculescu's free convolution operations. in 2022, i obtained a similar result for commutators of such random matrices; one feature of this result is the special role of even polynomials, in parallel with the situation in free probability.

it turns out that a certain family of special polynomials, called hypergeometric polynomials, arises naturally in relation to convolution of even polynomials and finite free commutators. i will explain how these polynomials can be used to approach questions of real-rootedness and asymptotics for finite free commutators. based on arxiv:2209.00523 and ongoing joint work with rafael morales and daniel perales.