Multilateral Inversion of A r , C r , and D r Basic Hypergeometric Series

In Electron. J. Combin. 10 (2003), #R10], the author presented a new basic hypergeometric matrix inverse with applications to bilateral basic hypergeometric series. This matrix inversion result was directly extracted from an instance of Bailey’s very-well-poised ₆ψ₆ summation theorem, and involves two infinite matrices which are not lower-triangular. The present paper features three different multivariable generalizations of the above result. These are extracted from Gustafson’s A _r and C _r extensions and from the author’s recent A _r extension of Bailey’s ₆ψ₆ summation formula. By combining these new multidimensional matrix inverses with A _r and D _r extensions of Jackson’s ₈ϕ₇ summation theorem three balanced verywell- poised ₈ψ₈ summation theorems associated to the root systems A _r and C _r are derived.