Michael J. Schlosser
Summation theorems for multidimensional basic hypergeometric series
by determinant evaluations
We derive summation formulas for a specific 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 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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