34.
The
C ? nonterminating C
? summation theorem is derived by appropriately specializing Gustafson's
6ψ
6 summation theorem for bilateral basic hypergeometric series very well-poised on symplectic
C ? groups. From this, the terminating
6?
5 and, hence, terminating
4?
3 summation theorem is obtained. A suitably modified
4?
3 is then used to derive the
C ? generalization of the Bailey transform. The transform is then interpreted as a matrix inversion result for two infinite, lower-triangular matrices. This result is used to motivate the definition of the
C ? Bailey pair. The
C ? generalization of Bailey's lemma is then proved. This result is inverted, and the concept of the bilateral Bailey chain is discussed. The
C ? Bailey lemma is then used to obtain a connection coefficient result for general
C ? little
q-Jacobi polynomials. All of this work is a natural extension of the unitary
A ?, or equivalently
U(?+1), case. The classical case, corresponding to
A 1 or equivalently
U(2), contains an immense amount of the theory and application of one-variable basic hypergeometric series, including elegant proofs of the Rogers-Ramanujan-Schur identities. The
C ? nonterminating
6?
5 summation theorem is also used to recover C. Krattenthaler's multivariable summation which he utilized in deriving his refinement of the Bender-Knuth and MacMahon generating functions for certain sets of plane partitions.
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