求导算子与恒等式

发布时间：2018-09-06 16:11

【摘要】：在这篇论文中,我在其他学术学者已给出的组合恒等式证明的基础上,利用部分分式法和算子方法等技巧,构造并计算得出了一些新的实用性强的组合恒等式,并在此基础上做了相应的推广和q-模拟。主要内容如下:1.主要利用部分分式法和高阶求导算子等数学工具,探讨(?)类型的调和数因式的结果。首先通过一些简单的因式了解了整个计算过程,然后得到了主要因式(?)的结果,作为Weideman恒等式的推广形式,对其参数取特殊值后,得到了许多漂亮的调和数恒等式。最后利用数学归纳法和Bell多项式验证了等式的存在性。2.首先运用Telescoping方法和算子方法构造了一个新的二项式系数恒等式:(?)对其进行代换和变形,得到了分子与分母互换的形式。然后分别利用一阶求导算子和二阶求导算子得到了上述调和数恒等式的一些推广。通过取特殊的参数值,得到了一些简单实用的调和数恒等式。3.通过对上述内容2的深入了解,发现其还有许多有趣之处,因此就对上述调和数恒等式以及2中得到的部分组合恒等式做了相应的q-模拟。
[Abstract]:In this paper, on the basis of the proof of combinatorial identities given by other scholars, some new combinatorial identities with strong practicability are constructed and calculated by using the techniques of partial fraction method and operator method. On this basis, some corresponding generalizations and q-simulations are made. The main contents are as follows: 1. The results of harmonic factors of type (?) are discussed with mathematical tools such as method and higher order derivative operator. First, the whole calculation process is understood through some simple factors, and then the results of main factors are obtained. As a generalization of Weideman identities, many beautiful harmonic identities are obtained by taking special values of their parameters. The existence of the equation is proved by mathematical induction and Bell polynomial. 2. A new binomial coefficient identity is constructed by using Telescoping method and operator method: (1) It is substituted and deformed, and the form of interchange between the denominator and the molecule is obtained. Some generalizations of the above-mentioned harmonic identities are given. Some simple and practical harmonic identities are obtained by taking special parameter values. 3. Through a thorough understanding of the above-mentioned content 2, we find that there are many interesting points. Therefore, we make corresponding q-simulation on the above-mentioned harmonic identities and some combinatorial identities obtained in 2.
【学位授予单位】：南京邮电大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157

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