Multiple orthogonal polynomials and branched continued fractions

The central theme of this talk is the recently found connection between two completely different corners of mathematics: multiple orthogonal polynomials and branched continued fractions introduced to solve total-positivity problems arising from combinatorics. Firstly, we give an overview of the connection between these two topics. Then, we give further evidence of this connection via the case study of the link between branched-continued-fraction representations for ratios of contiguous hypergeometric series and multiple orthogonal polynomials with respect to measures (or linear functionals) whose moments are ratios of products of Pochhammer symbols. Specialisations of these multiple orthogonal polynomials include the classical Laguerre, Jacobi, and Bessel orthogonal polynomials, multiple orthogonal polynomials with respect to two measures involving the Macdonald function, confluent hypergeometric functions, and Gauss’ hypergeometric function, and multiple orthogonal polynomials with respect to Meijer G-functions used to investigate the singular values of products of Ginibre random matrices. This is joint work with Alan Sokal.