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The hyperdeteminants considered here are the simplest analogues of determinants for higher rank tensors which have been defined by Cayley, and apply only to tensors with an even number of indices. We have shown in a previous article that the calculation of certain multidimensional integrals could be reduced to the evaluation of hyperdeterminants of Hankel type. Here, we carry out this computation by purely algebraic means in the cases of Selberg's and Aomoto's integrals.

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., Comment: 32 pages, LaTex, IOP macros