Summation theorems for multidimensional basic hypergeometric series by determinant evaluations Original Research Article

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 (one-dimensional) summation formulas with certain determinant evaluations. Our theorems include Ar extensions of Ramanujanʹs bilateral 1ψ1 sum, Cr extensions of Baileyʹs very-well-poised 6ψ6 summation, and a Cr extension of Jacksonʹs 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 Chuʹs 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 theorems of the classical theory of basic hypergeometric series.