# 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*\phi*7
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*\phi*9
transformation.
We obtain *C_r* and *D_r* 10*\phi*9 transformations from
an interchange of multisums, combined with *A_r*,
*C_r*, and *D_r* extensions of Jackson's 8*\phi*7 summation.
Special cases of our 10*\phi*9 transformations include
multivariable generalizations of Watson's transformation of an
8*\phi*7 into a multiple of a 4*\phi*3. We also deduce
multidimensional extensions of Sears' 4*\phi*3 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*\psi*1 sum, *C_r* extensions of Bailey's very-well-poised
6*\psi*6 summation, and a *C_r* extension of Jackson's
very-well-poised 8*\phi*7 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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