Consequences of the \(A_ l\) and \(C_ l\) Bailey transform and Bailey lemma
Over the past 15 years, a theory of identities for multiple (basic) hypergeometric series associated to root systems (but not requiring any knowledge of root systems) was built up, by Stephen Milne and his followers. This theory extends the classical theory of (basic) hypergeometric series form single sums to multiple summations. The paper under review constitutes a fundamental corner stone in the development of this theory. It is well-known that the classical Bailey lemma and Bailey transform allow to conveniently derive new identities from known ones. In this paper, two different extensions of the Bailey lemma and transform to the multiple case, one associated to root systems \(A_\ell\), the other associated to root systems \(C_\ell\), are given. Whereas previously proofs of identities in this multiple series theory required usually involved and labourious recursive arguments, these \(A_\ell\) and \(C_\ell \) extensions of the Bailey lemma and transform now allow to produce an host of new identities with ease, including extensions of many important classical identities to the multiple case. In this paper, \(A_\ell\) and \(C_\ell\) extensions of Watson's \(_8\Phi_7\)-to-\(_4\Phi_3\)-transformation, of Jackson's very-well-poised \(_8\Phi_7\)-summation, and of Sears' balanced \(_4\Phi_3\)-transformation are derived. Further applications can be found in [\textit{S. C. Milne}, ``Balanced \(_3\Phi_2\) summation theorems for \(U(n)\) basic hypergeometric series'' preprint; \textit{S. C. Milne} and \textit{J. W. Newcomb}, J. Comput. Appl. Math. 68, No. 1-2, 239-285 (1996; reviewed below)]. More general inversion formulas, which are at the heart of the Bailey lemma, were found by Bhatnagar, Milne, and Schlosser [\textit{G. Bhatnagar} and \textit{S. C. Milne}, ``Generalized bibasic hypergeometric series and their \(U(n)\) extension'', Adv. Math., to appear; \textit{M. Schlosser}, ``Multidimensional matrix inversions and \(A_r\) and \(D_r\) basic hypergeometric series'', Ramanujan J., to appear], and were applied in [M. Schlosser, ibid.] to find extensions of ``Gosper-type'' summation formulas, and in [G. Bhatnagar and M. Schlosser, ``\(C_n\) and \(D_n\) very-well-poised \(_{10} \Phi_9\) transformations'', preprint] to find new \(C_n\) and \(D_n\) extensions of many of the aforementioned classical identities.
scientific article; zbMATH DE number 773916
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