DX1(X1a2As...)

This article appears in Volume V01, Page 627 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: DRO-ECG

DX1(X1a2As...) _ (a2xa...),
while D,(ai)sa3..) =o unless the

partition

PARTITION

(81X283...) contains a

part

PART

s. Further, if DAiDA2 denote successive operations of DA1 and Da2, Da1Da2(X1a2a2...) _ (X3...),
and the operations are evidently commutative.
Also DPiDmDPB..(p;1p2ap33...) =1, and the law of operation of the operators D upon a monomial symmetric

function

FUNCTION

is clear. We have obtained the equivalent operations
1+D1+2D2 +3D3+... =exppol
where exp denotes (by the

rule

RULE

over exp) that the multiplication of operators is symbolic as in Taylor's theorem. d'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
a1 =a,+a1aa,+1+a2aaa}2+....
It has been shown (vide " Memoir on Symmetric Functions of the Roots of Systems of Equations," Phil. Trans. 189o, p. 490) that
exp(midi+m2d2+m3d3+...) =exp(M1d1+M2d2+Msda+...), where now the multiplications on the

dexter

denote successive operations, provided that
exp(M1f;+M22+M3r,3+...) =1+mlt+mzt;2+mat3+...; being an undetermined algebraic quantity.
Hence we derive the particular cases
expol = exp(di zd2+5(13 ...) ;
expadl = exp (dl -22d2+33d2 ...),
and we can express D. in terms of d1, d2, d3,..., products denoting successive operations, by the same law which expresses the elementary

function

FUNCTION

a, in terms of the sums of

powers

POWERS, HIRAM (1805-1873)

s1, S2, SS," Further, we can express da in terms of D1, D2, D2, ... by the same law which expresses the

power

POWER [WILLIAM GRATTAN] TYRONE (1797-1841)

function s, in terms of the elementary functions al, a2, a3,...
Operation of Da upon a Product of Symmetric Functions.Suppose f to be a product of symmetric functions flf2.. f . If in the identity f =f, f2...fm we introduce a new root we

change

CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

as into a1+/as1, and we obtain
(1 +D1+2D2+... +'Da+...)f
= (1-}-D1+112D2+.:.+'Da+...)f1
X (1 +D1+2D2+...+'Da+...) f2 X.
X (1+11D3+112D2++'Da+)fm,
and now expanding and equating coefficients of like

powers

POWERS, HIRAM (1805-1873)

of D1f =~(D1f1)f.fa...f
D2f =x(D2f1)f2f3...f +(Dlfl) (D1f2)f3... fm,
D3f =E(D3f1)f2f3...fm+ (D3f1)(Dlf2)f3...fm+(D3f1)f2fs...fm,
the summation in a

term

TERM

covering every distribution of the operators of the type presenting itself in the

term

TERM

.
Writing these results
D1f = D(nf,
D2f = D(2)f+Du2)f,
D3f = D<3)f+D(21)f+Da3)f,
et
B.,(ml,muemst, )
1, z, a,
we may write in general
De f =ED(plp2p3e.).f,
the summation being for every partition (plp2p3...) of s, and D(ptp2p3...)f being =E(Dpl.fl)(Dp2.f2)(DP3f3)f+...fm. Ex. gr. To operate with D2 upon (213)(214)(15), we have
D(2)fy' = (13) (Z14) (15) +(213) (14) (15).
Dc12).f = (122) (2P) (15) +(213) (213) (14) +(212) (214) (14),
and hence
D2f = (214) (15) (13) + (213) (15) (14) +(213) (212) (15) + (213)2 (14)
+(214) (212) (14).
Application.to Symmetric Function Multiplication.An example will explain this. Suppose we wish to find the coefficient of (52413) in the product (213)(214)(15).
Write
(213) (214) (15) _... +A(524) (13) +... ;
D,D2D7(213)(214)(15) =A;
every other term disappearing by the fundamental property of D,. Since
D,(213)(214)(15) =(13)(14)(14),
we have:

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