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Symmetry groups of Bailey’s transformations for \(_{10}\phi _{9}\)-series. (English) Zbl 1123.33014

Summary: Although most of the symmetry groups or “invariance groups” associated with two term transformations between (basic) hypergeometric series have been studied and identified, this is not the case for the most general transformation formulae in the theory of basic hypergeometric series, namely Bailey’s transformations for \(_{10}\phi _{9}\)-series. First, we show that the invariance group for both Bailey’s two term transformations for terminating \(_{10}\phi _{9}\)-series and Bailey’s four term transformations for non-terminating \(_{10}\phi _{9}\)-series (rewritten as a two term transformation of a so-called \(\Phi\)-series) is isomorphic to the Weyl group of type \(E_{6}\). We continue our recent research concerning the group structure underlying three term transformations [S. Lievens and J. Van der Jeugt, Invariance groups of three term transformations for basic hypergeometric series, J. Comput. Appl. Math. 197, 1–14 (2006; Zbl 1102.33015)] and demonstrate that the group associated with a three term transformation between these \(\Phi\)-series, each admitting Bailey’s two term transformation, is the Weyl group of type \(E_{7}\). We do this by giving a description of the root system of type \(E_{7}\) that allows to find a transformation between equivalent three term identities in an easy way. A computation shows that there are five, essentially different, three term transformations between these \(\Phi\)-series; we give an explicit form of each of these five transformations in an elegant way. To our knowledge only one of these transformations has appeared in the literature.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics

Citations:

Zbl 1102.33015

Software:

GAP; HYP
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Full Text: DOI

References:

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