q-Jacobi-Stirling数

发布时间：2018-01-11 00:02

本文关键词：q-Jacobi-Stirling数　出处：《大连海事大学》2017年硕士论文　论文类型：学位论文

【摘要】：Legendre-Stirling数是在Everitt探究经典二阶勒让德微分表达式的谱理论时提出来的,而且Legendre-Stirling数是拉格朗日对称式中勒让德表达式的积分复合幂的系数.Jacobi-Stirling数的概念是Everitt在2007年研究经典二阶雅各比微分表达式的谱理论时首次提出的.Jacobi-Stirling数是雅各比对称式中勒让德表达式的积分复合幂的系数.2012年,Mansour提出了q-类Stirling数并对其进行了相关的研究.由于 Jacobi-Stirling 数、Legendre-Stirling 数和类 Stirling 数与 Stirling 数有很多相似的性质,因此,受到很多的人关注.本文基于q-类Stirling数定义中基本函数及 Jacobi-Stirling 数和 Legendre-Stirling 数表达形式,提出了q-Jacobi-Stirling 数和q-Legendre-Stirling数的概念,并研究了其相关性质.本文的主要工作为以下三个方面:(1)通过基本函数[x]q=1-qx/1-q,引入新的"和函数"提出了两类q-Legendre-Stirling数、q-Jacobi-Stirling数的概念,推导出了它们满足的递推关系.(2)给出了第一类 q-Legendre-Stirling 数和第一类 q-Jacobi-Stirling 数的一种矩阵表示,证明了 q-Legendre-Stirling数和q-Jacobi-Stirling数的若干组合恒等式.(3)研究了两类q-Legendre-Stirling数之间的关系、两类q-Jacobi-Stirling数之间的关系,丰富了类Stirling数的研究成果.
[Abstract]:Legendre-Stirling number is proposed when Everitt explores the spectrum theory of classical Legendre differential expression. And the concept of Legendre-Stirling number is the coefficient of the integral compound power of Legendre expression in Lagrange symmetry, and the concept of Jacobi-Stirling number is Everi. The Jacobi-Stirling number, first proposed in 2007 when he studied the spectral theory of the classical second-order Yakubi differential expression, is the integral compound power of the Legendre expression in Yakubi's symmetric formula. Coefficient. 2012. Mansour put forward q-like Stirling number and studied it. Because of Jacobi-Stirling number. Legendre-Stirling numbers and similar Stirling numbers have many similar properties with Stirling numbers, so. This paper is based on the definition of q-class Stirling numbers and basic functions and Jacobi-Stirling numbers. Legendre-Stirling number expression. The concepts of q-Jacobi-Stirling number and q-Legendre-Stirling number are proposed. The main work of this paper is as follows: 1) pass through the basic function. [In this paper, the concept of q-Legendre-Stirling number and q-Jacobi-Stirling number is proposed by introducing a new "sum function". The recursion relation that they satisfy. A matrix representation of q-Legendre-Stirling numbers and q-Jacobi-Stirling numbers of the first kind is given. Some combinatorial identities of q-Legendre-Stirling number and q-Jacobi-Stirling number are proved. The relationship between two kinds of q-Legendre-Stirling numbers is studied. The relationship between two classes of q-Jacobi-Stirling numbers enriches the research results of the similar Stirling numbers.
【学位授予单位】：大连海事大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157

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