Hankel hyperdeterminants and Selberg integrals

TL;DR

This paper explores Hankel hyperdeterminants, generalizes classical properties, and establishes a connection with Selberg integrals, providing explicit evaluations and linking to physical models like Coulomb systems.

Contribution

It introduces a novel study of Hankel hyperdeterminants, generalizes classical determinant properties, and connects these to Selberg integrals and physical Coulomb systems.

Findings

Many Hankel hyperdeterminants can be evaluated explicitly.

Classical properties of Hankel determinants are extended to hyperdeterminants.

A link between hyperdeterminants and Selberg integrals is established.

Abstract

We investigate the simplest class of hyperdeterminants defined by Cayley in the case of Hankel hypermatrices (tensors of the form $A_{i_1i_2... i_k}=f(i_1+i_2+...+i_k)$). It is found that many classical properties of Hankel determinants can be generalized, and a connection with Selberg type integrals is established. In particular, Selberg's original formula amounts to the evaluation of all Hankel hyperdeterminants built from the moments of the Jacobi polynomials. Many higher-dimensional analogues of classical Hankel determinants are evaluated in closed form. The Toeplitz case is also briefly discussed. In physical terms, both cases are related to the partition functions of one-dimensional Coulomb systems with logarithmic potential.