Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/19

Elementary theory of Symmetric Functions

5

If we add all such functions which have the same weight we obtain, algebraically speaking, all the products ${\displaystyle w}$ together of the quantities ${\displaystyle \alpha ,\beta ,\gamma ,\ldots \nu }$, repetitions permissible.

Such a sum is called the Homogeneous Product-Sum of weight ${\displaystyle w}$ of the ${\displaystyle n}$ quantities.

It is usually denoted by ${\displaystyle h_{w}}$.

We have
{\displaystyle {\begin{aligned}\qquad \qquad h_{1}&=(1)=\textstyle \sum {\alpha }{\text{,}}\\\qquad \qquad h_{2}&=(2)+(1^{2})=\textstyle \sum {\alpha ^{2}}+\sum {\alpha \beta }{\text{,}}\\\qquad \qquad h_{3}&=(3)+(21)+(1^{3})=\textstyle \sum {\alpha ^{3}}+\sum {\alpha ^{2}\beta }+\sum {\alpha \beta \gamma }{\text{,}}\end{aligned}}}
and so forth.

We have before us the three sets of functions

{\displaystyle {\begin{aligned}&s_{1},s_{2},s_{3},\ldots s_{\nu },\ldots {\text{,}}\\&a_{1},a_{2},a_{3},\ldots a_{\nu }{\text{,}}\\&h_{1},h_{2},h_{3},\ldots h_{\nu },\ldots {\text{.}}\end{aligned}}}

The first and third sets contain an infinite number of members, but the second set only involves ${\displaystyle n}$ members where ${\displaystyle n}$ is the number of the quantities ${\displaystyle \alpha ,\beta ,\gamma ,\ldots }$.

6. The identity of Art. 4 which connects the functions ${\displaystyle a_{1},a_{2},a_{3},\ldots }$ with ${\displaystyle \alpha ,\beta ,\gamma ,\ldots }$ may be written, by putting ${\displaystyle {\tfrac {1}{y}}}$ for ${\displaystyle }$, ${\displaystyle 1-a_{1}y+a_{2}y^{2}-\ldots +(-)^{n}a_{n}y^{n}\equiv (1-\alpha y)(1-\beta y)\ldots (1-\nu y)}$, or in the form ${\displaystyle {\frac {1}{1-a_{1}y+a_{2}y^{2}-\ldots +(-)^{n}a_{n}y^{n}}}\equiv {\frac {1}{(1-\alpha y)(1-\beta y)\ldots (1-\nu y)}}}$. If we expand the last fraction in ascending powers of ${\displaystyle y}$, we obtain, in the first place, ${\displaystyle 1}$ ${\displaystyle +(\alpha +\beta +\gamma +\ldots +\nu )y}$ ${\displaystyle +(\alpha ^{2}\!+\beta ^{2}\!+\gamma ^{2}\!+\ldots +\nu ^{2}\!+\alpha \beta +\alpha \gamma +\beta \gamma +\ldots +\mu \nu )y^{2}}$ ${\displaystyle +(\alpha ^{3}\!\!\!+\!\beta ^{3}\!\!\!+\!\gamma ^{3}\!\!\!+\!\ldots \!+\!\nu ^{3}\!\!\!+\!\alpha ^{2}\!\beta \!+\!\alpha \beta ^{2}\!\!\!+\!\ldots \!+\!\mu ^{2}\!\nu \!+\!\mu \nu ^{2}\!\!\!+\!\alpha \beta \gamma \!+\!\alpha \beta \delta \!+\!\ldots \!+\!\!\lambda \mu \nu )y^{3}}$ ${\displaystyle +\ldots }$.
It is clear that the coefficient of ${\displaystyle y^{w}}$ is the homogeneous product-sum of weight ${\displaystyle w}$, so that we may write ${\displaystyle {\frac {1}{1-a_{1}y+a_{2}y^{2}-\ldots +(-)^{n}a_{n}y^{n}}}\equiv 1+h_{1}y+h_{2}y^{2}+\ldots +h_{w}y^{w}+\ldots }$ an identity.