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.

我们的第一个意图是验证 Forrester 猜想的 N_1=7 情况。我们试图证明，使用麦克唐纳微分算子，当用 Jack 多项式展开来自猜想的对称多项式时，某些部分不会出现。这在 N_1【小于或等于】6 的情况下当然成立，但是我们找不到获得必要数量的独立线性关系的展开系数来显示 的消失的一般过程。 2005年，我们主要研究了（非对称）Koornwinder多项式和与这些多项式相关的BC型双仿射Hecke代数，特别是研究了代数中广义对称算子和广义交替算子的性质。是这样的：在某些情况下推测并验证，如果将广义交替算子应用于 Koornwinder 多项式，则可以得到Koornwinder 多项式乘以关于部分变量的差积 该差积与 Baker-Forrester 猜想中出现的多项式完全相同（= Forrester 猜想的 q 模拟） 因此，Koornwinder 多项式的范数公式为：既然已经成立，我们可以预期该猜想是从范数公式得出的。

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