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626

ALGEBRAIC FORMS

( $l_{1}^{\lambda _{1}}l_{2}^{\lambda _{2}}...$ ) and it is seen intuitively that the number $\theta$ remains unaltered when the first two of these partitions are interchanged (see Combinatorial Analysis). Hence the theorem is established.

Putting $x_{1}=1$ and $x_{2}=x_{3}=x_{4}=...=0$ , we find a particular law of reciprocity given by Cayley and Betti,

$(1^{m_{1}})^{\mu _{1}}(1^{m_{2}})^{\mu _{2}}(1^{m_{3}})^{\mu _{3}}...=...+\theta (s_{1}^{\sigma _{1}}s_{2}^{\sigma _{2}}s_{3}^{\sigma _{3}}...)+...$ ,
$(1^{s_{1}})^{\sigma _{1}}(1^{s_{2}})^{\sigma _{2}}(1^{s_{3}})^{\sigma _{3}}...=...+\theta (m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}m_{3}^{\mu _{3}}...)+...$ ;

and another by putting $x_{1}=x_{2}=x_{3}=...=1$ , for then X_m becomes h_m, and we have

$h_{m_{1}}^{\mu _{1}}h_{m_{2}}^{\mu _{2}}h_{m_{3}}^{\mu _{3}}...=...+\theta ^{\prime }(s_{1}^{\sigma _{1}}s_{2}^{\sigma _{2}}s_{3}^{\sigma _{3}}...)+...$ ,
$h_{s_{1}}^{\sigma _{1}}h_{s_{2}}^{\sigma _{2}}h_{s_{3}}^{\sigma _{3}}...=...+\theta ^{\prime }(m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}m_{3}^{\mu _{3}}...)+...$ ,

Theorem of Expressibility.—"If a symmetric function be symboilized by $(\lambda \mu \nu ...)$ and $(\lambda _{1}\lambda _{2}\lambda _{3}...)$ , $(\mu _{1}\mu _{2}\mu _{3}...)$ , $(\nu _{1}\nu _{2}\nu _{3}...)$ ... be any partitions of $\lambda$ , $\mu$ , $\nu$ ... respectively, the function $(\lambda \mu \nu ...)$ is expressible by means of functions symbolized by separation of

$(\lambda _{1}\lambda _{2}\lambda _{3}...\mu _{1}\mu _{2}\mu _{3}...\nu _{1}\nu _{2}\nu _{3}...)$ ."

For, writing as before,

${\text{X}}_{m_{1}}^{\mu _{1}}{\text{X}}_{m_{2}}^{\mu _{2}}{\text{X}}_{m_{3}}^{\mu _{3}}...=\Sigma \Sigma \theta (s_{1}^{\sigma _{1}}s_{2}^{\sigma _{2}}s_{3}^{\sigma _{3}}...)x_{l_{1}}^{\lambda _{1}}x_{l_{2}}^{\lambda _{2}}x_{l_{3}}^{\lambda _{3}}...$ ,
⁠ $=\Sigma {\text{P}}x_{l_{1}}^{\lambda _{1}}x_{l_{2}}^{\lambda _{2}}x_{l_{3}}^{\lambda _{3}}...$ ,

P is a linear function of separations of $(l_{1}^{\lambda _{1}}l_{2}^{\lambda _{2}}l_{3}^{\lambda _{3}}...)$ of specification $(m_{1}^{\mu _{1}}m_{2}^{\mu _{2}}m_{3}^{\mu _{3}}...)$ , and if ${\text{X}}_{s_{1}}^{\sigma _{1}}{\text{X}}_{s_{2}}^{\sigma _{2}}{\text{X}}_{s_{3}}^{\sigma _{3}}...=\Sigma {\text{P}}^{\prime }x_{l_{1}}^{\lambda _{1}}x_{l_{2}}^{\lambda _{2}}x_{l_{3}}^{\lambda _{3}}...$ , ${\text{P}}^{\prime }$ is a linear function of separations of $(l_{1}^{\lambda _{1}}l_{2}^{\lambda _{2}}l_{3}^{\lambda _{3}}...)$ of specification $(s_{1}^{\sigma _{1}}s_{2}^{\sigma _{2}}s_{3}^{\sigma _{3}}...)$ . Suppose the separations of $(l_{1}^{\lambda _{1}}l_{2}^{\lambda _{2}}l_{3}^{\lambda _{3}}...)$ to involve $k$ different specifications and form the $k$ identities

${\text{X}}_{m_{1s}}^{\mu _{1s}}{\text{X}}_{m_{2s}}^{\mu _{2s}}{\text{X}}_{m_{3s}}^{\mu _{3s}}...=\Sigma {\text{P}}^{(s)}x_{l_{1}}^{\lambda _{1}}x_{l_{2}}^{\lambda _{2}}x_{l_{3}}^{\lambda _{3}}...(s=1,2,...k)$ ,

where $m_{1s}^{\mu _{1s}}m_{2s}^{\mu _{2s}}m_{3s}^{\mu _{3s}}...)$ is one of the $k$ specifications.

The law of reciprocity shows that

${\text{P}}^{(s)}=\sum _{t=1}^{t=k}\theta _{st}(m_{1l}^{\mu _{1l}}m_{2l}^{\mu _{2l}}m_{3l}^{\mu _{3l}}...)$ ,

viz.: a linear function of symmetric functions symbolized by the k specifications; and that $\theta _{st}=\theta _{ts}$ . A table may be formed expressing the k expressions ${\text{P}}^{(1)},{\text{P}}^{(2)},...{\text{P}}^{(k)}$ as linear functions of the k expressions $(m_{1l}^{\mu _{1l}}m_{2l}^{\mu _{2l}}m_{3l}^{\mu _{3l}}...)$ , s = 1, 2, ...k, and the numbers $\theta _{st}$ occurring therein possess row and column symmetry. By solving k linear equations we similarly express the latter functions as linear functions of the former, and this table will also be symmetrical.

Theorem.—“The symmetric function (mμ_1s
1s mμ_2s
2s mμ_3s
3s ....) whose partition is a specification of a separation of the function symbolized by (lλ₁
¹ lλ₂
² lλ₃
³...) is expressible as a linear function of symmetric functions symbolized by separations of (lλ₁
¹ lλ₂
² lλ₃
³...) and a symmetrical table may be thus formed.” It is now to be remarked that the partition (lλ₁
¹ lλ₂
² lλ₃
³...) can be derived from (mμ_1s
1s mμ_2s
2s mμ_3s
3s ....) by substituting for the numbers m_1s, m_2s, m_3s,... certain partitions of those numbers (vide the definition of the specification of a separation).

Hence the theorem of expressibility enunciated above. A new statement of the law of reciprocity can be arrived at as follows:—

Since.
⁠P^(s) = μ_1s! μ_2s! μ_3s! ... Σ (J₁)^j₁(J₂)^j₂ (J₃)^j₃.../j₁!j₂!j₃!...
where (J₁)^j₁(J₂)^j₂ (J₃)^j₃... is a separation of (lλ₁
¹ lλ₂
² lλ₃
³...) of specification (mμ_1s
1s mμ_2s
2s mμ_3s
3s ....), placing s under the summation sign to denote the specification involved,
⁠μ_1s! μ_2s! μ_3s! ... $\sum _{\textit {s}}$ (J₁)^j₁(J₂)^j₂ (J₃)^j₃.../j₁!j₂!j₃!... = $\sum _{\textit {t=1}}^{\textit {t=k}}$ θ_st (mμ_1t
1t mμ_2t
2t mμ_3t
3t ...)
⁠μ_1t! μ_2t! μ_3t! ... $\sum _{\textit {t}}$ (J₁)^j₁(J₂)^j₂ (J₃)^j₃.../j₁!j₂!j₃!... = $\sum _{\textit {s=1}}^{\textit {s=k}}$ θ_ts (mμ_1s
1s mμ_2s
2s mμ_3s
3s ...)
where θ_st = θ_ts.

Theorem of Symmetry.—If we form the separation function
⁠Σ (J₁)^j₁(J₂)^j₂ (J₃)^j₃.../j₁!j₂!j₃!...
appertaining to the function (lλ₁
¹ lλ₂
² lλ₃
³...), each separation having a specification (mμ_1s
1s mμ_2s
2s mμ_3s
3s ....), multiply by μ_1s! μ_2s! μ_3s! ... and take therein the coefficient of the function (mμ_1t
1t mμ_2t
2t mμ_3t
3t ....), we obtain the same result as if we formed the separation function in regard to the specification (mμ_1t
1t mμ_2t
2t mμ_3t
3t ....), multiplied by μ_1t! μ_2t! μ_3t! ... and took therein the coefficient of the function (mμ_1s
1s mμ_2s
2s mμ_3s
3s ....).

Ex.gr., take (l λ₁
¹ l λ₂
²...)=(21⁴);(mμ_1s
1s mμ_2s
2s...) = (321);(mμ_1t
1t mμ_2t
2t...)=(31₃); we find

(21)(1²)(1)+(1³)(2)(1)	=...+13(31³)+...,
(21)(1)³	= ...+13(321)+...

The Differential Operators.—Starting with the relation
⁠(1 + α₁x)(1 + α₂x)...(1 + α_nx) = 1+a₁x+a₂x²+...+a_nxⁿ
multiply each side by 1+μx, thus introducing a new quantity μ; we obtain
⁠(1 + α₁x)(1 + α₂x)...(1 + α_nx)(1+μx) = 1+(a₁+μ)x+(a₂+μa₁)x²+...
so that 𝑓(a₁, a₂, a₃,…a_n) = 𝑓, a rational integral function of the elementary functions, is converted into
⁠𝑓 (a₁+μ, a₂+μa₁,…a_n+μa_n−1) = 𝑓 + d₁ + μ²/2!d2
1𝑓 + μ³/3!d3
1𝑓 + …
where
⁠d₁=∂/∂a₁ + a₁∂/∂a₂ + a₂∂/∂a₃ + … + a_n−1∂/∂a_n
and ds
1 denotes, not s successive operations of d₁, but the operator of order s obtained by raising d₁ to the s^th power symbolically as in Taylor's theorem in the Differential Calculus.

Write also 1/s!ds
1 = D, so that
⁠𝑓 (a₁+μ, a₂+μa₁,…a_n+μa_n−1) = 𝑓 + μD₁𝑓 + μ²D₂𝑓 + μ³D₃𝑓 + …

The introduction of the quantity μ converts the symmetric function (λ₁λ₂λ₃…) into
⁠ (λ₁λ₂λ₃+…)+ μ^λ₁(λ₂λ₃…)+ μ^λ₂(λ₁λ₃…) + μ^λ₃(λ₁λ₂…)+…
Hence, if 𝑓 (a₁, a₂, a₃,…a_n) = (λ₁λ₂λ₃…),
⁠ (λ₁λ₂λ₃+…)+ μ^λ₁(λ₂λ₃…)+ μ^λ₂(λ₁λ₃…) + μ^λ₃(λ₁λ₂…)+…
⁠ = (1 + μD₁ + μ²D₂ + μ³D₃ + …)(λ₁λ₂λ₃…).

Comparing coefficients of like powers of μ we obtain
⁠Dλ₁(λ₁λ₂λ₃…) = (λ₂λ₃…),
while D_s(λ₁λ₂λ₃…) = 0 unless the partition (λ₁λ₂λ₃…) contains a part s. Further, if D_λ₁ D_λ₂ denote successive operations of D_λ₁ and D_λ₂,
⁠D_λ₁D_λ₂(λ₁λ₂λ₂…) = (λ₁…),
and the operations are evidently commutative.

Also Dπ₁
p₁Dπ₂
p₂Dπ₃
p₃ … (pπ₁
1pπ₂
2pπ₃
3 … ) = 1, and the law of operation of the operators D upon a monomial symmetric function is clear.

We have obtained the equivalent operations
⁠1 + μD₁ + μ²D₂ + μ³D₃ + … = expμd₁
where exp denotes (by the rule over exp) that the multiplication of operators is symbolic as in Taylor's theorem. ds
1 denotes, in fact, an operator of order s, but we may transform the right-hand side so that we are only concerned with the successive performance of linear operations. For this purpose write
⁠a_s = ∂_{a_s} + a₁∂_{a_s+1}+a₂∂_{a_s+2}+….

It has been shown (vide "Memoir on Symmetric Functions of the Roots of Systems of Equations," Phil. Trans. 1890, p. 490) that
⁠exp(m₁d₁ + m₂d₂ + m₃d₃ + …) = exp' (M₁d₁ + M₂d₂ + M₃d₃ + …),
where now the multiplications on the dexter denote successive operations, provided that
⁠exp(M₁ξ + M₂ξ² + M₃ξ³ + …) = 1 + m₁ξ + m₂ξ² + m₃ξ³ + ….
ξ being an undetermined algebraic quantity.

Hence we derive the particular cases
⁠expd₁ = exp(d₁ − 1/2d₂ + 1/3d₃ − …);
⁠expμd₁ = exp(μd₁ − 1/2μ²d₂ + 1/3μ³d₃ − …),
and we can express D_s in terms of d₁, d₂, d₃, …, products denoting successive operations, by the same law which expresses the elementary function a_s in terms of the sums of powers s₁, s₂, s₃, … Further, we can express d_s in terms of D₁, D₂, D₃, … by the same law which expresses the power function s, in terms of the elementary functions a₁, a₂, a₃, …

Operation of D_s a Product of Symmetric Functions.—Suppose 𝑓 to be a product of symmetric functions 𝑓₁𝑓₂…𝑓_m. If in the identity 𝑓 = 𝑓₁𝑓₂…𝑓_m we introduce a new root μ we change a_s into a_s + μa_s−1, and we obtain
⁠(1 + μD₁ + μ²D₂ + … + μ^sD_s + …) 𝑓
⁠=(1 + μD₁ + μ²D₂ + … + μ^sD_s + …) 𝑓₁
⁠×(1 + μD₁ + μ²D₂ + … + μ^sD_s + …) 𝑓₂
⁠×⁠⋅⁠⋅⁠⋅⁠⋅⁠⋅
⁠×(1 + μD₁ + μ²D₂ + … + μ^sD_s + …) 𝑓_m
and now expanding and equating coefficients of like powers of μ
⁠D₁𝑓 = Σ(D₁𝑓₁) 𝑓₂𝑓₃…𝑓_m,
⁠D₂𝑓 = Σ(D₂𝑓₁) 𝑓₂𝑓₃…𝑓_m + Σ(D₁𝑓₁)(D₁𝑓₂) 𝑓₃…𝑓_m,
⁠D₃𝑓 = Σ(D₃𝑓₁) 𝑓₂𝑓₃…𝑓_m + Σ(D₂𝑓₁)(D₁𝑓₂) 𝑓₃…𝑓_m + Σ(D₃𝑓₁) 𝑓₂𝑓₃…𝑓_m,
the summation in a term covering every distribution of the operators of the type presenting itself in the term.

Writing these results

${\text{D}}_{1}f={\text{D}}_{(1)}f$ ,
${\text{D}}_{2}f={\text{D}}_{(2)}f+{\text{D}}_{(1^{2})}f$ ,
${\text{D}}_{3}f={\text{D}}_{(3)}f+{\text{D}}_{(2^{1})}f+{\text{D}}_{(1^{2})}f$ ,

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