Multilateral Inversion of Ar, Cr, and Dr 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 Ar and Cr extensions and from the author’s recent Ar extension of Bailey’s 6ψ6 summation formula. By combining these new multidimensional matrix inverses with Ar and Dr extensions of Jackson’s 8ϕ7 summation theorem three balanced verywell- poised 8ψ8 summation theorems associated to the root systems Ar and Cr are derived.