Consequences of the Al and Cl Bailey transform and Bailey lemma Original Research Article

In this paper we study some applications of the higher-dimensional generalization of the Bailey transform, Bailey lemma, and iterative ‘Bailey chain’ concept in the setting of basic hypergeometric series very well-poised on unitary Al, or symplectic Cl, groups. The derivation of the Cl, case is closely related to the previous analysis of the unitary Al, case. Let G denote Al, or Cl. The G Bailey transform is obtained from a suitably modified G terminating very well-poised 4φ3 summation theorem and termwise transformations. It is then interpreted as a matrix inversion result for two infinite, lower-triangular matrices. This provides a higher-dimensional generalization of Andrewsʹ matrix inversion formulation of the Bailey transform. As in the classical case, the concept of a G Bailey pair is introduced, and then inverted. This G inversion applied to the G terminating very well-poised 6φ5 summations yields G terminating balanced 3φ2 summations. The G Bailey lemma is obtained directly from a G terminating very well-poised 6φ5 summation theorem and the matrix inversion formulation of the G Bailey transform. It shows how to construct another G Bailey pair from an arbitrary G Bailey pair. The concepts of an ordinary G Bailey chain and a bilateral G Bailey chain are introduced. Finally, as an example, we give one Al, and one Cl, q-Whipple transformation, and some of their applications. These include G, q-Dougall summations and G 4φ3 sears transformations. The classical case of all this work, corresponding to Al or equivalently U (2), contains an immense amount of the theory and application of one-variable basic hypergeometric series.