Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/21

Rh and as shewn in works upon algebra

where $$\pi_1 !$$ denotes the factorial of $$\pi_1$$ and

the summation being taken for all sets of positive integers $$\pi_1, \pi_2, \ldots, \pi_k$$ which satisfy this equation.

By interchange of symbols we pass to the relations
 * $$\begin{align}

a_1&=h_1\text{,}\\ a_2&={h_1}^2-h_2\text{,}\\ a_3&={h_1}^3-2h_1h_2+h_3\text{,}\\ \ldots&\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\ a_n&={\textstyle\sum{}}(-)^{n+\pi_1+\pi_2+\ldots+\pi_k} \frac{(\pi_1+\pi_2+\ldots+\pi_k)!}{\pi_1!\,\pi_2!\ldots\pi_k!} {h_1}^{\pi_1}{h_2}^{\pi_2}\ldots{h_k}^{\pi_k}\text{.} \end{align}$$

8. It is shewn in works upon algebra that the relations between the symbols $$s_1, s_2, s_3, \ldots$$ and the symbols $$a_1, a_2, a_3, \ldots$$ are
 * $$\begin{align}

s_1&=a_1\text{,}\\ s_2&={a_1}^2-2a_2\text{,}\\ s_3&={a_1}^3-3a_1h_2+3a_3\text{,}\\ \ldots&\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\ s_n&={\textstyle\sum{}}(-)^{n+\pi_1+\pi_2+\ldots+\pi_k} \frac{(\pi_1+\pi_2+\ldots+\pi_k-1)!\,n}{\pi_1!\,\pi_2!\ldots\pi_k!} {a_1}^{\pi_1}{a_2}^{\pi_2}\ldots{a_k}^{\pi_k}\text{.} \end{align}$$ $$\begin{align} a_1&=s_1\text{,}\\ 2!\,a_2&={s_1}^2-s_2\text{,}\\ 3!\,a_3&={s_1}^3-3s_1s_2+2s_3\text{,}\\ \ldots&\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\ n!\,a_n&={\textstyle\sum{}}(-)^{n+\pi_1+\pi_2+\ldots+\pi_k} \frac{n!}{1^{\pi_1}.2^{\pi_2}\ldots k^{\pi_k}.\pi_1!\,\pi_2!\ldots\pi_k!} {s_1}^{\pi_1}{s_2}^{\pi_2}\ldots{s_k}^{\pi_k}\text{;} \end{align}$$ also between the symbols $$s_1, s_2, s_3, \ldots$$ and $$h_1, h_2, h_3, \ldots$$
 * $$\begin{align}

s_1&=h_1\text{,}\\ s_2&=-({h_1}^2-2h_2)\text{,}\\ s_3&={h_1}^3-3h_1h_2+3h_3\text{,}\\ \ldots&\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\ s_n&={\textstyle\sum{}}(-)^{\pi_1+\pi_2+\ldots+\pi_k+1} \frac{(\pi_1+\pi_2+\ldots+\pi_k-1)!\,n}{\pi_1!\,\pi_2!\ldots\pi_k!} {h_1}^{\pi_1}{h_2}^{\pi_2}\ldots{h_k}^{\pi_k}\text{.} \end{align}$$