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 A_r, C_r, and D_r. Our new summation formulas include D_r Jackson's 8\phi7 summations, A_r and D_r quadratic, and D_r cubic summations. Further, we derive multivariable generalizations of Bailey's classical terminating balanced very-well-poised 10\phi9 transformation. We obtain C_r and D_r 10\phi9 transformations from an interchange of multisums, combined with A_r, C_r, and D_r extensions of Jackson's 8\phi7 summation. Special cases of our 10\phi9 transformations include multivariable generalizations of Watson's transformation of an 8\phi7 into a multiple of a 4\phi3. We also deduce multidimensional extensions of Sears' 4\phi3 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 A_r extensions of Ramanujan's bilateral 1\psi1 sum, C_r extensions of Bailey's very-well-poised 6\psi6 summation, and a C_r extension of Jackson's very-well-poised 8\phi7 summation formula. We also derive multidimensional extensions, associated to the classical root systems of type A_r, B_r, C_r, and D_r, 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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