#### Title of Presentation

Girard-Waring Identities and Their Applications

#### Type of Submission

Event

#### Faculty Advisor

Tian-Xiao He

#### Graduation Year

2020

#### Location

Center for Natural Sciences, Illinois Wesleyan University

#### Start Date

21-4-2018 9:00 AM

#### End Date

21-4-2018 10:00 AM

#### Disciplines

Education

#### Abstract

Our research project is about application of recursive sequences in the construction of a class of combinatorial identities called Girard-Waring identities. This type of identities is derived from recursive sequences, which is the motivation and the guiding light of our path to deeper understanding of mathematics. A sequence constructed from a recessive relation is called recursive sequence, which starts from a few initial quantities to generate a sequence of quantities by using a simple relationship in modeling some real world problems or mathematical problems. As a natural math model of those problems, recursive sequences are an important tool widely used in Combinatorics and Graph Theory, Number Theory, Fractal, Cryptography, etc. Many identities in elementary mathematics and other advanced mathematics come from the Girard-Waring identities. We connected the generating function of a linear recursive sequence and its explicit expression to give an efficient method to construct Girard-Waring type identities. We also used the method in the study of some construction problems such as summation formulas, Hagen-Rothe type identities, etc. In addition, some applications of those summation formulas and identities are discussed.

Girard-Waring Identities and Their Applications

Center for Natural Sciences, Illinois Wesleyan University

Our research project is about application of recursive sequences in the construction of a class of combinatorial identities called Girard-Waring identities. This type of identities is derived from recursive sequences, which is the motivation and the guiding light of our path to deeper understanding of mathematics. A sequence constructed from a recessive relation is called recursive sequence, which starts from a few initial quantities to generate a sequence of quantities by using a simple relationship in modeling some real world problems or mathematical problems. As a natural math model of those problems, recursive sequences are an important tool widely used in Combinatorics and Graph Theory, Number Theory, Fractal, Cryptography, etc. Many identities in elementary mathematics and other advanced mathematics come from the Girard-Waring identities. We connected the generating function of a linear recursive sequence and its explicit expression to give an efficient method to construct Girard-Waring type identities. We also used the method in the study of some construction problems such as summation formulas, Hagen-Rothe type identities, etc. In addition, some applications of those summation formulas and identities are discussed.