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

Rh 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 $$\begin{align} \qquad\qquad h_1&=(1)=\textstyle\sum{\alpha}\text{,}\\ \qquad\qquad h_2&=(2)+(1^2)=\textstyle\sum{\alpha^2}+\sum{\alpha\beta}\text{,}\\ \qquad\qquad h_3&= (3) + (21) + (1^3) =\textstyle\sum{\alpha^3}+\sum{\alpha^2\beta}+\sum{\alpha\beta\gamma}\text{,} \end{align} $$ and so forth.

We have before us the three sets of functions $$\begin{align} &s_1, s_2, s_3, \ldots s_\nu, \ldots\text{,}\\ &a_1, a_2, a_3, \ldots a_\nu\text{,}\\ &h_1, h_2, h_3, \ldots h_\nu, \ldots\text{.}\end{align}$$

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 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 $$x$$,

or in the form

If we expand the last fraction in ascending powers of $$y$$, we obtain, in the first place,

$$+\ldots$$. It is clear that the coefficient of $$y^w$$ is the homogeneous product-sum of weight $$w$$, so that we may write

an identity.