Abstract
TheC ℓ nonterminating Cℓ summation theorem is derived by appropriately specializing Gustafson's6ψ6 summation theorem for bilateral basic hypergeometric series very well-poised on symplecticC ℓ groups. From this, the terminating6ϕ5 and, hence, terminating4ϕ3 summation theorem is obtained. A suitably modified4ϕ3 is then used to derive theC ℓ 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 theC ℓ Bailey pair. TheC ℓ generalization of Bailey's lemma is then proved. This result is inverted, and the concept of the bilateral Bailey chain is discussed. TheC ℓ Bailey lemma is then used to obtain a connection coefficient result for generalC ℓ littleq-Jacobi polynomials. All of this work is a natural extension of the unitaryA ℓ, or equivalentlyU(ℓ+1), case. The classical case, corresponding toA 1 or equivalentlyU(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. TheC ℓ nonterminating6ϕ5 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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Communicated by Tom Koornwinder.
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Lilly, G.M., Milne, S.C. TheC ℓ Bailey transform and Bailey lemma. Constr. Approx 9, 473–500 (1993). https://doi.org/10.1007/BF01204652
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DOI: https://doi.org/10.1007/BF01204652