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

and as shewn in works upon algebra $h_{n}={\textstyle \sum {}}(-)^{n+\pi _{1}+\pi _{2}+\ldots +\pi _{k}}{\frac {(\pi _{1}+\pi _{2}+\ldots +\pi _{k})!}{\pi _{1}!\,\pi _{2}!\ldots \pi _{k}!}}{a_{1}}^{\pi _{1}}{a_{2}}^{\pi _{2}}\ldots {a_{k}}^{\pi _{k}}$, where $\pi _{1}!$ denotes the factorial of $\pi _{1}$ and $\pi _{1}+2\pi _{2}+3\pi _{3}+\ldots +k\pi _{k}=n$, 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

${\begin{aligned}a_{1}&=h_{1}{\text{,}}\\a_{2}&={h_{1}}^{2}-h_{2}{\text{,}}\\a_{3}&={h_{1}}^{3}-2h_{1}h_{2}+h_{3}{\text{,}}\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\a_{n}&={\textstyle \sum {}}(-)^{n+\pi _{1}+\pi _{2}+\ldots +\pi _{k}}{\frac {(\pi _{1}+\pi _{2}+\ldots +\pi _{k})!}{\pi _{1}!\,\pi _{2}!\ldots \pi _{k}!}}{h_{1}}^{\pi _{1}}{h_{2}}^{\pi _{2}}\ldots {h_{k}}^{\pi _{k}}{\text{.}}\end{aligned}}$

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

${\begin{aligned}s_{1}&=a_{1}{\text{,}}\\s_{2}&={a_{1}}^{2}-2a_{2}{\text{,}}\\s_{3}&={a_{1}}^{3}-3a_{1}h_{2}+3a_{3}{\text{,}}\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\s_{n}&={\textstyle \sum {}}(-)^{n+\pi _{1}+\pi _{2}+\ldots +\pi _{k}}{\frac {(\pi _{1}+\pi _{2}+\ldots +\pi _{k}-1)!\,n}{\pi _{1}!\,\pi _{2}!\ldots \pi _{k}!}}{a_{1}}^{\pi _{1}}{a_{2}}^{\pi _{2}}\ldots {a_{k}}^{\pi _{k}}{\text{.}}\end{aligned}}$

${\begin{aligned}a_{1}&=s_{1}{\text{,}}\\2!\,a_{2}&={s_{1}}^{2}-s_{2}{\text{,}}\\3!\,a_{3}&={s_{1}}^{3}-3s_{1}s_{2}+2s_{3}{\text{,}}\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\n!\,a_{n}&={\textstyle \sum {}}(-)^{n+\pi _{1}+\pi _{2}+\ldots +\pi _{k}}{\frac {n!}{1^{\pi _{1}}.2^{\pi _{2}}\ldots k^{\pi _{k}}.\pi _{1}!\,\pi _{2}!\ldots \pi _{k}!}}{s_{1}}^{\pi _{1}}{s_{2}}^{\pi _{2}}\ldots {s_{k}}^{\pi _{k}}{\text{;}}\end{aligned}}$
also between the symbols $s_{1},s_{2},s_{3},\ldots$ and $h_{1},h_{2},h_{3},\ldots$

${\begin{aligned}s_{1}&=h_{1}{\text{,}}\\s_{2}&=-({h_{1}}^{2}-2h_{2}){\text{,}}\\s_{3}&={h_{1}}^{3}-3h_{1}h_{2}+3h_{3}{\text{,}}\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\s_{n}&={\textstyle \sum {}}(-)^{\pi _{1}+\pi _{2}+\ldots +\pi _{k}+1}{\frac {(\pi _{1}+\pi _{2}+\ldots +\pi _{k}-1)!\,n}{\pi _{1}!\,\pi _{2}!\ldots \pi _{k}!}}{h_{1}}^{\pi _{1}}{h_{2}}^{\pi _{2}}\ldots {h_{k}}^{\pi _{k}}{\text{.}}\end{aligned}}$