广西师范大学学报（自然科学版） ›› 2015, Vol. 33 ›› Issue (4): 73-80.doi: 10.16088/j.issn.1001-6600.2015.04.013

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Dunkl-Clifford分析框架下的Hermite多项式

李珊珊1, 费铭岗2   

  1. 1.西南民族大学计算机科学与技术学院,四川成都610041;
    2.电子科技大学数学科学学院,四川成都611731
  • 收稿日期:2015-03-16 出版日期:2015-12-25 发布日期:2018-09-21
  • 通讯作者: 李珊珊(1980—),女,四川成都人,西南民族大学讲师。E-mail: ssrubyli@163.com
  • 基金资助:
    国家自然科学基金资助项目(11301054);四川省应用基础计划资助项目(2013JY0180);西南民族大学中央高校基金资助项目(2015NZYQN27)

Hermite Polynomials Related to the Dunkl-Clifford Analysis

LI Shan-shan1, FEI Ming-gang2   

  1. 1. College of Computer Science and Technology, Southwest University for Nationalities,Chengdu Sicuan 610041, China;
    2. School of Mathematical Sciences, University of Electronic Scienceand Technology of China, Chengdu Sicuan 611731, China
  • Received:2015-03-16 Online:2015-12-25 Published:2018-09-21

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摘要： 本文从纯分析的角度出发,利用Dunkl-Dirac算子的球坐标表示,得到了Dunkl-Clifford分析框架下关于Dunkl算子任意正整数次幂,尤其是奇数次幂下经典Hermite多项式的推广形式。并且作为应用,本文建立了Dunkl-Clifford分析中Hermite多项式所满足的微分方程。

关键词: 反射群, Dunkl-Dirac算子, Hermite多项式

Abstract: In this paper, based on the classical method from Clifford analysis and a spherical representation of Dunkl-Dirac operator, a generalization of the classical Hermite polynomials related to the framework of Dunkl operators is presented. For application, the associated differential equation about Hermite polynomials in Dunkl-Clifford analysis setting is established.

Key words: reflection group, Dunkl-Dirac operator, Hermite polynomials

中图分类号: 

  • O174.22
[1] DUNKL C F. Differential-difference operators associated to reflection groups[J]. Trans Amer Math Soc, 1989, 311(1): 167-183.
[2] DUNKL C F, XU Yuan. Orthogonal polynomials of several variables[M]. Cambridge: Cambridge University Press, 2001.
[3] RÖSLER M. Dunkl Operators: Theory and applications[M]//Orthogonal Polynomials and Special Functions: Lecture Notes in Mathematics Volume 1817. Berlin: Springer, 2003:93-135.
[4] FEI Ming-gang, CEREJEIRAS P, KÄHLER U. Fueter’s theorem and its generalizations in Dunkl-Clifford analysis[J]. J Phys A: Math Theor, 2009, 42(39): 395209.
[5] RÖSLER M. Generalized Hermite polynomials and the heat equation for Dunkl operators[J]. Comm Math Phys, 1998, 192(3): 519-542.
[6] FEI Ming-gang, CEREJEIRAS P, KÄHLER U. Spherical Dunkl-Monogenics and a factorization of the Dunkl-Laplacian[J]. J Phys A: Math Theor, 2010, 43(44): 445202.
[7] 费铭岗. 迭代的Dunkl-Dirac算子的基本解及其应用[J]. 数学物理学报, 2013,33A(6): 1052-1061.
[8] SOMMEN F. Special functions in Clifford analysis and axial symmetry[J]. J Math Anal Appl, 1988, 130(1): 110-133.
[9] DE BIE H. An alternative definition of the Hermite polynomials related to the Dunkl Laplacian[J]. SIGMA, 2008, 4:093.
[10] DE BIE H, SOMMEN F. Hermite and Gegenbauer polynomials in superspace using Clifford analysis[J]. J Phys A: Math Theor, 2007, 40(34): 10441.
[11] DE BIE H,ØRSTED B, SOMBERG P, et al. Dunkl operators and a family of realizations of osp(1|2)}[J]. Trans Amer Math Soc, 2012, 364(7), 3875-3902.
[12] DELANGHE R, SOMMEN F, SOUCEK V. Clifford algebra and spinor-valued functions: a function theory for Dirac operator[M]. Berlin: Springer, 1992.
[13] 乔玉英,黄沙,赵红芳,等. Clifford分析中一类非线性边值问题[J]. 数学物理学报,1996,16(3):284-290.
[14] 黄沙,焦红兵,乔玉英,等. Clifford分析中多个未知函数向量的非线性边值问题[J]. 数学学报,1998,41(6):1185-1192.
[15] 乔玉英. 双正则函数的非线性带位移边值问题[J]. 系统科学与数学,1999,19(4):484-489.
[16] 杨丕文. k-正则函数及某些边值问题[J]. 四川师范大学学报:自然科学版,2001,24(1):5-8.
[17] 谢永红,黄沙,乔玉英. 广义双正则函数带共轭值带位移的边值问题[J]. 数学研究与评论,2001,21(4):567-572.
[18] 乔玉英,王丽丽,焦红兵. 实Clifford分析中的拟Bochner-Martinelli型高阶奇异积分[J]. 数学物理学报,2005,25A(3):289-298.
[19] 赵丽琴,乔玉英,李尊凤,等. 实Clifford分析中正则函数列的几条性质[J]. 数学的实践与认识,2005,35(2):177-182.
[20] CEREJEIRAS P, KÄHLER U, REN Guang-bin. Clifford analysis for finite reflection groups[J]. Complex Var Elliptic Equ, 2006, 51(5/6): 487-495.
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