Michael Schlosser

Multidimensional matrix inversions and multiple basic hypergeometric series

(114 pages)

Abstract. We compute the inverse of a specific infinite r-dimensional matrix, thus unifying multidimensional matrix inversions recently found by Milne, Lilly, and Bhatnagar. Our inversion is an r-dimensional extension of a matrix inversion previously found by Krattenthaler. We also compute the inverse of another infinite r-dimensional matrix. As applications of our matrix inversions, we derive new summation formulas for multidimensional basic hypergeometric series. We work in the setting of multiple basic hypergeometric series very-well-poised on the root systems Ar, Cr, and Dr. Our new summation formulas include Dr Jackson's 8φ7 summations, Ar and Dr quadratic, and Dr cubic summations. Further, we derive multivariable generalizations of Bailey's classical terminating balanced very-well-poised 10φ9 transformation. We obtain Cr and Dr 10φ9 transformations from an interchange of multisums, combined with Ar, Cr, and Dr extensions of Jackson's 8φ7 summation. Special cases of our 10φ9 transformations include multivariable generalizations of Watson's transformation of an 8φ7 into a multiple of a 4φ3. We also deduce multidimensional extensions of Sears' 4φ3 transformation. Furthermore, we derive summation formulas for a different kind of multidimensional basic hypergeometric series associated to root systems of classical type. We proceed by combining the classical one-dimensional summation formulas with certain determinant evaluations. Our theorems include Ar extensions of Ramanujan's bilateral 1ψ1 sum, Cr extensions of Bailey's very-well-poised 6ψ6 summation, and a Cr extension of Jackson's very-well-poised 8φ7 summation formula. We also derive multidimensional extensions, associated to the classical root systems of type Ar, Br, Cr, and Dr, respectively, of Chu's bilateral transformation formula for basic hypergeometric series of Gasper-Karlsson-Minton type. Limiting cases of our various series identities include multidimensional generalizations of many of the most important summation and transformation theorems of the classical theory of basic hypergeometric series.

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