Hankel hyperdeterminants and Selberg integrals

Details Hankel hyperdeterminants and Selberg integrals Hankel hyperdeterminants and Selberg integrals

2002-11-19

arxiv.org/abs/math-ph/0211044v1

Physics

Mathematical Physics

32 pages, LaTex, IOP macros

Scientific paper

10.1088/0305-4470/36/19/306

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.

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