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Elliptic hypergeometric series on root systems. (English) Zbl 1066.33017
The author studies elliptic hypergeometric series on root systems. He obtains elliptic analogues for the Gustafson--Milne type series on the root systems $A_n$, $C_n$ and $D_n$. In the case of root systems $A_n$ and $D_n$ the proof is based on an elliptic partial fraction expansion and an induction. For this root system $C_n$ the multivariable hypergeometric summations are obtained through convenient determinant evaluations. The main formula is deduced as a special case of a multivariable Jackson sum of {\it S. O. Warnaar} [Constructive Approximation, 18, 479-502 (2002; Zbl 1040.33013)]. From these summation and transformation formulas there are deduced corresponding elliptic Bailey transformations.

##### MSC:
 33D67 Basic hypergeometric functions associated with root systems 11F50 Jacobi forms
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##### References:
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