Hyperdeterminantal expressions for Jack functions of rectangular shapes

We derive a Jacobi–Trudi type formula for Jack functions of rectangular shapes. In this formula, we make use of a hyperdeterminant, which is Cayleyʹs simple generalization of the determinant. In addition, after developing the general theory of hyperdeterminants, we give summation formulas for Schur functions involving hyperdeterminants, and evaluate Toeplitz type hyperdeterminants by using Jack function theory.