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

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

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

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

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

We have before us the three sets of functions

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

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

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