Date of Graduation

1996

Document Type

Dissertation/Thesis

Abstract

Gould (The Fibonacci Quarterly 2, (1964), 241-60) proved the general inversion theorem;{dollar}{dollar}f(n,k)=\\sum\\limits\\sbsp{lcub}j=k{rcub}{lcub}n{rcub} g(n,j)R(j,k)\\ {lcub}\\rm if\\ and\\ only\\ if{rcub}\\ g(n,k)=\\sum\\limits\\sbsp{lcub}j=k{rcub}{lcub}n{rcub}\\ f(n,j)A(j,k){dollar}{dollar}for any ordered sequence pair (f(n,k),g(n,k)), where R(n,k) is the number of compositions of {dollar}n\\ge 1{dollar} into k relatively prime parts and {dollar}A(n,k){dollar} is its inverse. In chapter one we obtain a variety of such ordered inversion pairs (f(n,k),g(n,k)) and give necessary and sufficient conditions for the congruence f(n,k) {dollar}\\equiv{dollar} g(n,k) (mod k) to hold, in particular criteria for {dollar}k\\ge 2{dollar} to be a prime when the congruence holds for all {dollar}n\\ge 1.{dollar} For example the pair{dollar}{dollar}(f(n,k),g(n,k))=(\\bigl({lcub}n\\atop k{rcub}\\bigr)\\left({lcub}{lcub}2n-k{rcub}\\atop n{rcub}\\right),\\ \\sum\\limits\\sbsp{lcub}i=k{rcub}{lcub}n{rcub}\\left\\lbrack{lcub}i\\over k{rcub}\\right\\rbrack\\bigl{lcub}n\\atop k{rcub}\\bigr)\\sp2){dollar}{dollar}and the corresponding congruence{dollar}{dollar}\\left({lcub}n\\atop k{rcub}\\right)\\left({lcub}{lcub}2n-k{rcub}\\atop n{rcub}\\right)\\equiv\\sum\\limits\\sbsp{lcub}i=k{rcub}{lcub}n{rcub}\\left\\lbrack {lcub}i\\over k{rcub}\\right\\rbrack\\left({lcub}n\\atop i{rcub}\\right)\\sp2\\ ({lcub}\\rm mod{rcub}\\ k),{dollar}{dollar}for all {dollar}n\\ge 1{dollar} and {dollar}k\\ge 2{dollar} if and only if k is a prime number. Gould's work was inspired by the congruence {dollar}\\bigl({lcub}n\\atop k{rcub}\\bigr)\\equiv\\bigl\\lbrack{lcub}n\\over k{rcub}\\bigr\\rbrack{dollar} (mod k), for all {dollar}n\\ge 1{dollar} and {dollar}k\\ge 2{dollar} if and only if k is a prime number. In chapter two we investigate the function {dollar}\\gamma(n)=\\sum\\limits\\sbsp{lcub}k=1{rcub}{lcub}n{rcub}\\lbrack k,n\\rbrack{dollar} and its generalizations. An identity involving this function and the function{dollar}{dollar}\\beta(n)=\\sum\\limits\\sbsp{lcub}k=1{rcub}{lcub}n{rcub} (k,n){dollar}{dollar}is established. We also obtain results on the genralized Euler phi function{dollar}{dollar}\\eqalign{lcub}\\phi\\sb{lcub}f{rcub}(n,m,a)=&\\sum\\quad\\quad f(k),\\cr {lcub}1\\le &k\\le n{rcub}\\atop{lcub}(k,a)=m{rcub}\\cr{rcub}{dollar}{dollar}and the function{dollar}{dollar}\\eqalign{lcub}T\\sb{lcub}m{rcub}(n)=&\\sum\\quad\\quad\\quad\\quad\\cr 1{lcub}\\le a\\sb1,a\\sb2,&{lcub}\\cdots{rcub},a\\sb{lcub}m{rcub}\\le n{rcub}\\atop{lcub}\\lbrack a\\sb1,a\\sb2,{lcub}\\cdots{rcub},a\\sb{lcub}m{rcub}\\rbrack=n{rcub}\\cr{rcub}{dollar}{dollar}a generalization of Cesaro's function{dollar}{dollar}\\eqalign{lcub}&\\sum\\quad\\quad 1=\au(n\\sp2).\\cr {lcub}1\\le a&,b\\le n{rcub}\\atop{lcub}\\lbrack a,b\\rbrack=n{rcub}\\cr{rcub}{dollar}{dollar}.

Download

Share

COinS