Elementary theory of Symmetric Functions
7
and as shewn in works upon algebra
,
where
denotes the factorial of
and
,
the summation being taken for all sets of positive integers
which satisfy this equation.
By interchange of symbols we pass to the relations
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b288b226c4c2168fa3904f33f71fc3549541ce)
8. It is shewn in works upon algebra that the relations between the
symbols
and the symbols
are
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c2ebdf3e536911f5535ac7efab46ed97ada98c4)
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1f9d8d38b929919fb99ef215736424314892c96)
also between the symbols
and
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63dfee2f1f3bcd98be8acec4128d28236c1d4d0d)