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 6-psi-6 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 of the author's recent A_r extension of Bailey's 6-psi-6 summation formula. By combining these new multidimensional matrix inverses with A_r and D_r extensions of Jackson's 8-phi-7 summation theorem three balanced very-well-poised 8-psi-8 summation theorems associated with the root systems A_r and C_r are derived.Comment: 24 page

A new multivariable 6-psi-6 summation formula

By multidimensional matrix inversion, combined with an A_r extension of Jackson's 8-phi-7 summation formula by Milne, a new multivariable 8-phi-7 summation is derived. By a polynomial argument this 8-phi-7 summation is transformed to another multivariable 8-phi-7 summation which, by taking a suitable limit, is reduced to a new multivariable extension of the nonterminating 6-phi-5 summation. The latter is then extended, by analytic continuation, to a new multivariable extension of Bailey's very-well-poised 6-psi-6 summation formula.Comment: 16 page