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Multilateral Inversion of Ar, Cr, and Dr Basic Hypergeometric Series
Michael J. Schlosser
Fakult\"at f\"ur Mathematik, Universit\"at Wien, Nordbergstra{\ss}e 15, A-1090 Vienna, Austria
Annals of Combinatorics 13 (3) pp.341-363 September, 2009
AMS Subject Classification: 33D67; 15A09, 33D15
In [Electron. J. Combin. \textbf{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 from 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 to the root systems $A_r$ and $C_r$ are derived.
Keywords: Bilateral basic hypergeometric series, Bailey's ${}_6\psi_6$ summation, Jackson's ${}_8\phi_7$ summation, ${}_8\psi_8$ summation, $A_r$ series, $C_r$ series, $D_r$ series


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