Page:EB1911 - Volume 01.djvu/668

and we see further that $(\sigma _{1}a_{1}+\sigma _{2}a_{2}+\ldots +\sigma _{m}a_{m})^{k}$ vanishes identically unless $k=0({\text{mod }}m)$. If $m$ be infinite and

$$1+b_{1}x+b_{2}x^{2}+\ldots =(1+\sigma _{1}x)(1+\sigma _{2}x)\ldots =e^{\beta _{1^{x}}}=e^{\beta _{2^{x}}}=\ldots ,$$

we have the symbolic identity

$$e^{\sigma _{1}a_{1}+\sigma _{2}a_{2}+\sigma _{3}a_{3}+\ldots }=e^{\rho _{1}\beta _{1}+\rho _{2}\beta _{2}+\rho _{3}\beta _{3}+\ldots ,}$$

and

$$(\sigma _{1}a_{1}+\sigma _{2}a_{2}+\sigma _{3}a_{3}+\ldots )^{p}=(\rho _{1}\beta _{1}+\rho _{2}\beta _{2}+\rho _{3}\beta _{3}+\ldots )^{p}.$$

Instead of the above symbols we may use equivalent differential operators. Thus let

$$\delta _{a}=a_{1}\delta _{a_{0}}+2a_{2}\delta _{a_{1}}+3a_{3}\delta _{a_{2}}+\ldots$$

and let $a,b,c,\ldots$ be equivalent quantities. Any function of differences of $\delta _{a},\delta _{b},\delta _{c},\ldots$ being formed, the expansion being carried out, an operand $a_{0}$ or $b_{0}$ or $c_{0}\ldots$ being taken and $b,c,\ldots$ being subsequently put equal to $a$, a non-unitary symmetric function will be produced.

Ex. gr.

$${\begin{aligned}(&\delta _{a}-\delta _{b})^{2}(\delta _{a}-\delta _{c})=(\delta _{a}^{2}-2\delta _{a}\delta _{b}+\delta _{b}^{2})(\delta _{a}-\delta _{c})\\&=\delta _{a}^{3}-2\delta _{a}^{2}\delta _{b}+\delta _{a}\delta _{b}^{2}-\delta _{a}^{2}\delta _{c}+2\delta _{a}\delta _{b}\delta _{c}-\delta _{b}^{2}\delta _{c}\\&=6a_{3}-4a_{2}b_{1}+2a_{1}b_{2}-2a_{2}c_{1}+2a_{1}b_{1}c_{1}-2b_{2}c_{1}\\&=2(a_{1}^{3}-3a_{1}a_{2}+3a_{3})=2(3).\end{aligned}}$$

The whole theory of these forms is consequently contained implicitly in the operation $\delta .$

Symmetric Functions of Several Systems of Quantities.—It will suffice to consider two systems of quantities as the corresponding theory for three or more systems is obtainable by an obvious enlargement of the nomenclature and notation.

Taking the systems of quantities to be

$$a_{1},a_{2},a_{3},\ldots$$

$$\beta _{1},\beta _{2},\beta _{3},\ldots$$

we start with the fundamental relation

$${\begin{aligned}&(1+a_{1}x+\beta _{1}y)(1+a_{2}x+\beta _{2}y)(1+a_{3}x+\beta _{3}y)\ldots \\=\,&1+a_{10}x+a_{01}y+a_{20}x^{2}+a_{11}xy+a_{02}y^{2}+\ldots +a_{pq}x^{p}y^{q}+\ldots \end{aligned}}$$

As shown by L. Schläfli^[1] this equation may be directly formed and exhibited as the resultant of two given equations, and an arbitrary linear non-homogeneous equation in two variables. The right-hand side may be also written

$$1+\Sigma a_{1}x+\Sigma \beta _{1}y+\Sigma a_{1}a_{2}x^{2}+\Sigma a_{1}\beta _{2}xy+\Sigma \beta _{1}\beta _{2}y^{2}+\ldots$$

The most general symmetric function to be considered is

$$\Sigma a_{1}^{p_{1}}\beta _{1}^{q_{1}}a_{2}^{p_{2}}\beta _{2}^{q_{2}}a_{3}^{p_{3}}\beta _{3}^{q_{3}}\ldots$$

conveniently written in the symbolic form

$$({\overline {p_{1}q_{1}}}{\overline {p_{2}q_{2}}}{\overline {p_{3}q_{3}}}\ldots ).$$

Observe that the summation is in regard to the expressions obtained by permuting the $n$ suffixes $1,2,3,\ldots n.$ The weight of the function is bipartite and consists of the two numbers $\Sigma p$ and $\Sigma q;$ the symbolic expression of the symmetric function is a partition into biparts (multiparts) of the bipartite (multipartite) number ${\overline {\Sigma p,\Sigma q}}.$ Each part of the partition is a bipartite number, and in representing the partition it is convenient to indicate repetitions of parts by power symbols. In this notation the fundamental relation is written

$${\begin{aligned}(&1+a_{1}x+\beta _{1}y)(1+a_{2}x+\beta _{2}y)(1+a_{3}x+\beta _{3}y)\ldots \\=1&+({\overline {10}})x+({\overline {01}})y+({\overline {10}}^{2})x^{2}+({\overline {10}}\,{\overline {01}})xy+({\overline {01}}^{2})y^{2}\\&+({\overline {10}}^{3})x^{3}+({\overline {10}}^{2}{\overline {01}})x^{2}y+({\overline {10}}\,{\overline {01}}^{2})xy^{2}+({\overline {01}}^{3})y^{3}+\ldots \end{aligned}}$$

where in general $a_{pq}=({\overline {10}}^{p}{\overline {01}}^{q}).$

All symmetric functions are expressible in terms of the quantities $a_{pq}$ in a rational integral form; from this property they are termed elementary functions; further they are said to be single-unitary since each part of the partition denoting $a_{pq}$ involves but a single unit.

The number of partitions of a biweight ${\overline {pq}}$ into exactly $\mu$ biparts is given (after Euler) by the coefficient of $a\mu x^{p}y^{q}$ in the expansion of the generating function

$${\frac {1}{1-ax.\,1-ay.\,1-ax^{2}.\,1-axy.\,1-ay^{2}.\,1-ax^{3}.\,1-ax^{2}y.\,1-axy^{2}.\,1-ay^{3}\ldots }}$$

The partitions with one bipart correspond to the sums of powers in the single system or unipartite theory; they are readily expressed in terms of the elementary functions. For write $({\overline {pq}})=s_{pq}$ and take logarithms of both sides of the fundamental relation; we obtain

$$s_{10}x+s_{01}y=\Sigma (a_{1}x+\beta _{1}y)$$

$$s_{20}x^{2}+2s_{11}xy+s_{02}y^{2}=\Sigma (a_{1}x+\beta _{1}y)^{2},\,\&{\text{c}}.,$$

and

$$s_{10}x+s_{01}y-{\frac {1}{2}}(s_{20}x^{2}+2s_{11}xy+s_{02}y^{2})+\ldots$$

$$=\log {(1+a_{10}x+a_{01}y+\ldots +a_{pq}x^{p}y^{q}+\ldots )}$$

From this formula we obtain by elementary algebra

$$(-)^{p+q-1}{\frac {(p+q-1)!}{p!\,q!}}s_{pq}=\sum _{\pi }(-)^{\Sigma \pi -1}{\frac {(\Sigma \pi -1)!}{\pi _{1}!\,\pi _{2}!\ldots }}a_{p_{1}q_{1}}^{\pi _{1}}a_{p_{2}q_{2}}^{\pi _{2}}\ldots$$

corresponding to Thomas Waring’s formula for the single system. The analogoous formula appertaining to $n$ systems of quantities which express $s_{pqr\ldots }$ in terms of elementary functions can be at once written down.

Ex. gr. ⁠ We can verify the relations

$${\begin{aligned}&s_{30}=a_{1\,0}^{3}-3a_{20}a_{10}+3a_{30},\\&s_{21}=a_{1\,0}^{2}a_{01}-a_{20}a_{01}-a_{11}a_{10}+a_{21}.\end{aligned}}$$

The formula actually gives the expression of $({\overline {pq}})$ by means of separations of

$$({\overline {10^{p}}}\,{\overline {01^{q})}},$$

which is one of the partitions of $({\overline {pq}}).$ This is the true standpoint from which the theorem should be regarded. It is but a particular case of a general theory of expressibility.

To invert the formula we may write

$$1+a_{10}x+a_{01}y+\ldots +a_{pq}x^{p}y^{q}+\ldots$$

$$=exp\;\{(s_{10}x+s_{01}y)-{\frac {1}{2}}s_{20}x^{2}+2s_{11}xy+s_{02}y^{2})+\ldots \},$$

and thence derive the formula—

$$(-)^{p+q-1}a_{pq}$$

$$=\sum \;{\bigg \{}{\frac {(p_{1}+q_{1}-1)!}{p_{1}!q_{1}!}}{\bigg \}}^{\pi _{1}}{\bigg \{}{\frac {(p_{2}+q_{2}-1)!}{p_{2}!q_{2}!}}{\bigg \}}^{\pi _{2}}\ldots {\frac {(-)^{\Sigma \pi -1}}{\pi _{1}!\,\pi _{2}!\ldots }}s_{p_{1}q_{1}}^{\pi _{1}}s_{p_{2}q_{2}}^{\pi _{2}}\ldots ,$$

which expresses the elementary function in terms of the single bipart functions. The similar theorem for $n$ systems of quantities can be at once written down.

It will be shown later that every rational integral symmetric function is similarly expressible.

The Function $h_{pq}$.—As the definition of $h_{pq}$ we take

$$1+n_{10}x+n_{01}y+\ldots +n_{pq}x^{p}y^{q}+\ldots$$

$$={\frac {1}{(1-a_{1}x-\beta _{1}y)(1-a_{2}x-\beta _{2}y)\ldots }};$$

and now expanding the right-hand side

$$h_{pq}=\sum {\bigg (}{\frac {p_{1}+q_{1}}{p_{1}}}{\bigg )}{\bigg (}{\frac {p_{2}+q_{2}}{p_{2}}}{\bigg )}\ldots ({\overline {p_{1}q_{1}}}\,{\overline {p_{2}q_{2}}}\ldots ),$$

the summation being for all partitions of the biweight. Further writing

$$1+h_{10}x+h_{01}y+\ldots +h_{pq}x^{p}y^{q}+\ldots$$

$$={\frac {1}{1-a_{10}x-a_{01}y+\ldots +(-)^{p+q}a_{pq}x^{p}y^{q}+\ldots }},$$

we find that the effect of changing the signs of both $x$ and $y$ is merely to interchange the symbols $a$ and $h;$ hence in any relation connecting the quantities $h_{pq}$ with the quantities $a_{pq}$ we are at liberty to interchange the symbols $a$ and $h.$ By the exponential and multinomial theorems we obtain the results—

$$(-)^{p+q-1}h_{pq}=\sum _{\pi }(-)^{\Sigma \pi -1}{\frac {(\Sigma \pi )!}{\pi _{1}!\,\pi _{2}!\ldots }}h_{p_{1}q_{1}}^{\pi _{1}}h_{p_{2}q_{2}}^{\pi _{2}}\ldots ;$$

$$h_{pq}=\sum {\bigg \{}{\frac {(p_{1}+q_{1}-1)!}{p_{1}!\,q_{1}!}}{\bigg \}}^{\pi _{1}}{\bigg \{}{\frac {(p_{2}+q_{2}-1)!}{p_{2}!\,q_{2}!\ldots }}{\bigg \}}^{\pi _{2}}\ldots {\frac {1}{\pi _{1}!\,\pi _{2}!\ldots }}s_{\underline {p_{1}q_{1}}}^{\pi _{1}}s_{p_{2}q_{2}}^{\pi _{2}}\ldots .$$

Differential Operations.—If, in the identity

$${\begin{aligned}&(1+a_{1}x+\beta _{1}y)(1+a_{2}x+\beta _{2}y)\ldots (1+a_{n}x+\beta _{n}y)\\=\,&1+a_{10}x+a_{01}y+a_{20}x^{2}+a_{11}xy+a_{02}y^{2}+\ldots ,\end{aligned}}$$

we multiply each side by $(1+\mu x+\nu y),$ the right-hand side becomes

$$1+(a_{10}+\mu )x+(a_{01}+\nu )y+\ldots +(a_{pq}+\mu a_{p-1,q}+\nu a_{p,q-1})x^{p}y^{q}+\ldots ;$$

hence any rational integral function of the coefficients $a_{10},a_{01},\ldots a_{pq},\ldots$ say $f(a_{10},a_{01},\ldots )\equiv f$ is converted into

$${\overline {exp}}(\mu d_{10}+\nu d_{01})f$$

$${\text{where }}d_{10}=\sum a_{p-1,q}{\frac {d}{da_{pq}}},d_{01}=\sum a_{p,q-1}{\frac {d}{da_{pq}}}.$$

The rule over $exp$ will serve to denote that $\mu d_{10}+\nu d_{01}$ is to be raised to the various powers symbolically as in Taylor’s theorem.

Writing

$${\text{D}}_{pq}={\frac {1}{p!\,q!}}d_{10}^{p}d_{01}^{q},$$

$${\overline {exp}}(\mu d_{10}+\nu d_{01})=(1+\mu {\text{D}}_{10}+\nu {\text{D}}_{01}+\ldots +\mu ^{p}\nu ^{q}{\text{D}}_{pq}+\ldots )f;$$

now, since the introduction of the new quantities $\mu ,\nu$ results in the addition to the function $({\overline {p_{1}q_{1}}}\,{\overline {p_{2}q_{2}}}\,{\overline {p_{3}q_{3}}}\ldots )$ of the new terms

$$\mu ^{p_{1}}\nu ^{q_{1}}({\overline {p_{2}q_{2}}}\,{\overline {p_{3}q_{3}}}\ldots )+\mu ^{p_{2}}\nu ^{q_{2}}({\overline {p_{1}q_{1}}}\,{\overline {p_{3}q_{3}}}\ldots )+\mu ^{p_{3}}\nu ^{q_{3}}({\overline {p_{1}q_{1}}}\,{\overline {p_{2}q_{2}}}\ldots )+\ldots ,$$

we find

$${\text{D}}_{p_{1}q_{1}}({\overline {p_{1}q_{1}}}\,{\overline {p_{2}q_{2}}}\,{\overline {p_{3}q_{3}}}\ldots )=({\overline {p_{2}q_{2}}}\,{\overline {p_{3}q_{3}}}\ldots );$$

and thence

$${\text{D}}_{p_{1}q_{1}}{\text{D}}_{p_{2}q_{2}}{\text{D}}_{p_{3}q_{3}}\ldots ({\overline {p_{1}q_{1}}}\,{\overline {p_{2}q_{2}}}\,{\overline {p_{3}q_{3}}}\ldots )=1;$$

while ${\text{D}}_{rs}f=0$ unless the part ${\overline {rs}}$ is involved in $f.$ We may then state that ${\text{D}}_{pq}$ is an operation which obliterates one part ${\overline {pq}}$ when such part is present, but in the contrary case causes the function to

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1. Vienna Transactions, t. iv. 1852.