A Proof of a Multivariable Elliptic Summation Formula Conjectured by Warnaar
نویسنده
چکیده
Elliptic hypergeometric series form a natural generalization of hypergeometric and basic hypergeometric (or q-) series. It is surprising that they were introduced only very recently, by Frenkel and Turaev [FT], who expressed the 6j-symbols corresponding to certain elliptic solutions of the Yang–Baxter equation, cf. [DJ], in terms of the 10ω9-sums defined below. It is expected that elliptic hypergeometric series play a fundamental role in the representation theory of elliptic quantum groups, though so far there has been little work in this direction. Recall that a series ∑
منابع مشابه
PROOF OF A SUMMATION FORMULA FOR AN Ãn BASIC HYPERGEOMETRIC SERIES CONJECTURED BY WARNAAR
Abstract. A proof of an unusual summation formula for a basic hypergeometric series associated to the affine root system Ãn that was conjectured byWarnaar is given. It makes use of Milne’s An extension of Watson’s transformation, Ramanujan’s 1ψ1summation, and a determinant evaluation of the author. In addition, a transformation formula between basic hypergeometric series associated to the affin...
متن کاملGustafson–Rakha-Type Elliptic Hypergeometric Series
We prove a multivariable elliptic extension of Jackson’s summation formula conjectured by Spiridonov. The trigonometric limit case of this result is due to Gustafson and Rakha. As applications, we obtain two further multivariable elliptic Jackson summations and two multivariable elliptic Bailey transformations. The latter four results are all new even in the trigonometric case.
متن کاملOn Warnaar’s Elliptic Matrix Inversion and Karlsson–minton-type Elliptic Hypergeometric Series
Using Krattenthaler’s operator method, we give a new proof of Warnaar’s recent elliptic extension of Krattenthaler’s matrix inversion. Further, using a theta function identity closely related to Warnaar’s inversion, we derive summation and transformation formulas for elliptic hypergeometric series of Karlsson–Minton-type. A special case yields a particular summation that was used by Warnaar to ...
متن کاملTheta Functions, Elliptic Hypergeometric Series, and Kawanaka’s Macdonald Polynomial Conjecture
We give a new theta-function identity, a special case of which is utilised to prove Kawanaka’s Macdonald polynomial conjecture. The theta-function identity further yields a transformation formula for multivariable elliptic hypergeometric series which appears to be new even in the one-variable, basic case.
متن کاملAn Elementary Proof of Ramanujan’s Circular Summation Formula and Its Generalizations
In this paper, we give a completely elementary proof of Ramanujan’s circular summation formula of theta functions and its generalizations given by S. H. Chan and Z. -G. Liu, who used the theory of elliptic functions. In contrast to all other proofs, our proofs are elementary. An application of this summation formula is given.
متن کامل