A determinant formula for a holonomic q-difference system associated with Jackson integrals of type BCn

Abstract

A Jackson integral of type BC n is a multisum generalization of the very-well-poised-balanced 2r ψ 2r 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.

Connection Formula for the Jackson Integral of Type A_n and Elliptic Lagrange Interpolation

“…where L = (l λµ ) λ,µ∈Zs,n and U = (u λµ ) λ,µ∈Zs,n are lower and upper triangular matrices of size s+n−1 n with respect to the lexicographic ordering , respectively. The diagonal entries of L and U are given by 2) and the determinant of the coefficient matrix is…”

A First Order q-Difference System for the BC1-Type Jackson Integral and Its Applications

“…Remark 4.1. The q-difference system for the BC n -type Jackson integral is discussed in [3] for its rank, and in [4,5] for the explicit expression of the determinant of the coefficient matrix of the system. On the other hand, though it is only for the BC 1 -type Jackson integral, the coefficient matrix in its Gauss decomposition form is obtained explicitly only in the present paper.…”

“…Remark 4.2. For further results on BC n -type Jackson integrals, see [2,4,3,5,13,14], for instance. Now we state the recurrence relation for Gustafson's sum 1, ξ G .…”