Proof of a summation formula for an $\tilde A_n$ basic hypergeometric series conjectured by Warnaar
Christian Krattenthaler
备注：
9 pages, AmS-LaTeX
发表信息：
in: "q-Series with Applications to Combinatorics, Number Theory, and Physics," Urbana-Champaign, Oct. 26-28, 2000, B. C. Berndt, K. Ono, eds., Contemporary Math., vol. 291, Amer. Math. Soc., Providence, R.I., 2001, pp. 153-161
A proof of an unusual summation formula for a basic hypergeometric series associated to the affine root system $\tilde A_n$ that was conjectured by Warnaar is given. It makes use of Milne's $A_n$ extension of Watson's transformation, Ramanujan's $_1ψ_1$-summation, and a determinant evaluation of the author. In addition, a transformation formula between basic hypergeometric series associated to the affine root systems $\tilde A_n$ respectively $\tilde A_m$, which generalizes at the same time the above summation formula and an identity due to Gessel and the author, is proposed as a conjecture.
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 derive quadratic, cubic and quartic transformations for elliptic hypergeometric series. Starting from another theta function identity, we derive yet different summation and transformation formulas for el…
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