Pairs of Comparable Relations for Complete Homogeneous Symmetric Polynomial

Soumendra Bera

doi:10.5539/jmr.v7n4p26

Abstract

<p>Complete homogeneous symmetric polynomial has connections with binomial coefficient, composition, elementary symmetric polynomial, exponential function, falling factorial, generating series, odd prime and Stirling numbers of the second kind by different summations. Surprisingly the relations in the context are comparable in pairs. </p>

Highlights

For each nonnegative integer n, complete homogeneous symmetric polynomial hn(x1, ... , xk ) or in brief hn{xk} is the sum of all distinct monomials of degree n in the variables: x1, ... , xk
The monomials again belong to the distinct symmetric polynomials
A recurrence function for the polynomial, which is analogous with a function for falling factorial, helps to find the relation of the polynomial with elementary symmetric polynomial

Summary

Introduction

For each nonnegative integer n, complete homogeneous symmetric polynomial hn(x1, ... , xk ) or in brief hn{xk} is the sum of all distinct monomials of degree n in the variables: x1, ... , xk. Xk ) or in brief hn{xk} is the sum of all distinct monomials of degree n in the variables: x1, ... Hn{xk} is the sum of all distinct monomial symmetric polynomials of degree n in k variables: x1, ... ){xk} = Distinct monomial symmetric polynomial of k variables: x1, ... From the formal definition of hn{xk}, it follows that a term of hn + 1{yk} is a term of hn + 1{xk + 1}, which does not contain xm as a factor; and if xm is multiplied with a term of hn{xk + 1} the product is a term of hn + 1{xk + 1} , which contains xm as a factor This implies that hn + 1{yk} is the sum of some terms of hn + 1{xk + 1}. We show other applications of (1.1) in the subsequent topics

Kind 1 with a Series of k Terms

Pair of Expressions for