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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 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