Menon Identities On The Algebraic Integer Ring And Others

Let K be an algebraic number field and OK the ring of algebraic integers in K.This paper studies two identities on OK,which is Menon identity and Cesaro formula.In 1965,Menon proved Menon identity.In 2012,Tarnauceanu extended the Menon identity to the upper triangle matrix.In 2017,Li-Kim fixed the result of Tarnauceanu,and gave some new identities.Inspired by Tarnauceanu,Li-Kim,we consider the matrix group acts on OK.By the Cauchy-Frobenius-Burnside lemma,we find and prove two Menon-type identities on OK.In 1885,Cesaro found the Cesaro formula over Z.In 2017,Miguel obtained the Cesaro formula over OK.On the basis of them,we extend the Cesaro formula on Z and OK to m variables.The general framework of the paper is as follows:In Chapter 1,we introduce the background and development of Menon identity and Cesaro formula,at the same time,we give some main results of this paper.In Chapter 2,we review some preliminary knowledge needed for the article,including some basic definitions and necessary lemmas.In Chapter 3,by choosing two special matrix groups and Cauchy-Frobenius-Burnside lemma,we get two new Menon-type identities.In Chapter 4,we study Cesaro formula and generalize it to m variables.