Forrester's conjecture and a generalization of Selberg intengral

福雷斯特猜想和塞尔伯格积分的推广

• 批准号: 16540198
• 负责人: KANEKO Jyoichi
• 金额: $ 1.02万
• 依托单位: The University of the Ryukyus
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540198/
• 关键词: Forrester's conjecture; Jack polynomial; Macdonald polynomial; Koornwinder polynomial; double affine Hecke algebra; Norm formula; Selberg型積分; 一般化Jacobi多項式

It was our first intension to verify the N_1=7 case of Forrester's conjecture. We tried to show that, using Macdonald differential operators, when one expands a symmetric polynomilal that comes from the conjecture in terms of Jack polynomials, certain part of terms does not appear. This certainly holds in the cases N_1【less than or equal】6, but we could not find the general procedure to get necessary number of independent linear relations of expansion coefficients to show vanishing of certain terms.In this year of 2005, we studied mainly (nonsymmetric) Koornwinder polynomials and the double affine Hecke algebra of type BC associated to these polynomials. Especially we have investigated properties of generalized symmetrization operators and generalized alternating operators in the algebra. The reason is this : It is conjectured and verified in some cases that if one applies the generalized alternating operator to a Koornwinder polynomial, then one obtains a Koornwinder polynomial multiplied by a difference product with respect to part of variables. This difference product is exactly the same polynomial appearing in the conjecture of Baker-Forrester (=q-analogue of Forrester's conjecture). Hence, as the norm formula of Koornwinder polynomials is already established, we may expect that the conjecture follows from the norm formula.

这是我们第一个验证Forrester猜想的N_1 = 7情况的目的。我们试图证明，使用MacDonald差异操作员，当人们扩展了从插孔多项式方面来自协议的对称多项式时，就不会出现某些术语的一部分。在n_1【少于或相等的情况下，这肯定存在，但是我们找不到一般的程序来获得必要数量的扩展的独立线性关系，以显示某些术语的消失。特别是我们研究了代数中广义对称运算符和广义替代操作员的特性。原因是：在某些情况下，它是猜想和验证的，如果将广义的替代操作员应用于Koornwinder多项式，则获得了koornwinder多项式相对于变量的一部分，将koornwinder多项式乘以差异产品。该差异产物与贝克 - 福雷斯特（Baker-Forterster）概念（= forrester的猜想的Q-Analeogue）完全相同。因此，由于已经建立了Koornwinder多项式的规范公式，因此我们可能希望该概念来自规范公式。

项目成果

On an extremal problem of Selberg

关于 Selberg 的极值问题

• 发表时间: 2006
• 期刊: Journal of Approximation Theory 142
• 作者: K. Kuroki;et al.;金子譲一;金子譲一;今野均;今野 均;金子 譲一
• 通讯作者: 金子 譲一

Connection formulas for algebraic hypergeometric functions

代数超几何函数的连接公式

• 发表时间: 2005
• 期刊: Kyushu J.of Math. 59
• 作者: Tomida S;Mamiya T;Abe K;Yoshimura T;Nabeshima T;Ebihara S.;M.Kato
• 通讯作者: M.Kato

Generalized hypergeometric functions _nF_<n-1> with monodromy group S_<n+1>

具有单数群 S_<n 1> 的广义超几何函数 _nF_<n-1>

• 期刊: Funkcialaj Ekvacioj (発表予定)
• 作者: Shimomura;S.;Jyoichi Kaneko;池田 薫;Kaoru Ikeda;Mitsuo Kato;Kaoru Ikeda;Jyoichi Kaneko;Mitsuo Kato;Mitsuo Kato;Mitsuo Kato;Mitsuo Kato;Mitsuo Kato;Mitsuo Kato;Mitsuo Kato
• 通讯作者: Mitsuo Kato