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A Jackson integral of type BC n is a multisum generalization of the very-well-poised-balanced ψ 2 r 2 r basic hypergeometric series. We state an explicit product formula for the determinant of a matrix with entries given by the BC n type Jackson integrals. In order to show this, we treat the determinant as a solution of a holonomic q-difference equation. In particular we give the q-difference equation explicitly as a two-term recurrence relation, which the determinant satisfies, by introducing a set of new symmetric polynomials via the symplectic Schur functions.

We study a q-difference equation of a BCn-type Jackson integral introduced by van Diejen. The BCn-type Jackson integral is a multiple q-series generalized from a q-analogue of Selberg's integral. The equation is characterized by some new symmetric polynomials defined via symplectic Schur functions. As an application, we give another proof of a product formula for the BCn-type Jackson integral, which is equivalent to the so-called q-Macdonald?Morris identity for the root system BCn first obtained by Gustafson.