MULTILATERAL INVERSION OF A_r, C_r AND D_r BASIC HYPERGEOMETRIC SERIES

MULTILATERAL INVERSION OF A_r, C_r AND D_r BASIC HYPERGEOMETRIC SERIES

30 Aug 2006 | MICHAEL J. SCHLOSSER
This paper presents three multivariable generalizations of a matrix inversion result originally derived from Bailey’s very-well-poised $_6\psi_6$ summation theorem. These generalizations are extracted from Gustafson’s $A_r$ and $C_r$ extensions and 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. The paper also includes a detailed introduction to basic hypergeometric series and multidimensional basic hypergeometric series associated with root systems, as well as several multi-sum identities. The proofs of the main theorems are provided, and the paper concludes with an appendix that discusses an incorrect application of multidimensional inverse relations leading to a divergent $D_r$ very-well-poised $_6\psi_6$ summation.This paper presents three multivariable generalizations of a matrix inversion result originally derived from Bailey’s very-well-poised $_6\psi_6$ summation theorem. These generalizations are extracted from Gustafson’s $A_r$ and $C_r$ extensions and 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. The paper also includes a detailed introduction to basic hypergeometric series and multidimensional basic hypergeometric series associated with root systems, as well as several multi-sum identities. The proofs of the main theorems are provided, and the paper concludes with an appendix that discusses an incorrect application of multidimensional inverse relations leading to a divergent $D_r$ very-well-poised $_6\psi_6$ summation.
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[slides and audio] Multilateral Inversion of Ar%2C Cr%2C and Dr Basic Hypergeometric Series