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Symmetry groups underlying Bailey's transfomations for $^{10}\phi^9$ series

Tuesday 12th September 2006 - 11:30 to 12:00
INI Seminar Room 1

The concept of symmetry groups associated with two term transformations for basic hypergeometric series is well known, and most of them have been studied and identified (J. Math. Phys. 1999:6692+ and references therein). One two term identity for which the invariance group, to our knowledge, was not written down explicitly is Bailey's four term transformation for non-terminating ${}_{10}\phi_9$-series considered as a two term transformation between a linear combination of such series which we call $\Phi$. It is shown that the invariance group of this transformation is the Weyl group of type $E_6$.

We demonstrate that the group associated with a three term transformation between $\Phi$-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. We also show how one can find a prototype of each of the five essentially different three term identities between $\Phi$-series.

Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons