Multidimensional matrix inversions and multiple basic hypergeometric series
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
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
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
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.
The following versions are available:
Back to Christian Krattenthaler's