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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