Page:EB1911 - Volume 01.djvu/666

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626
ALGEBRAIC FORMS

and it is seen intuitively that the number remains unaltered when the first two of these partitions are interchanged (see Combinatorial Analysis). Hence the theorem is established.

Putting and we find a particular law of reciprocity given by Cayley and Betti,

and another by putting , for then becomes , and we have

Theorem of Expressibility.—“If a symmetric function be symboilized by and be any partitions of respectively, the function is expressible by means of functions symbolized by separation of

For, writing as before,

is a linear function of separations of of specification and if is a linear function of separations of of specification Suppose the separations of to involve different specifications and form the identities

where is one of the specifications.

The law of reciprocity shows that

viz.: a linear function of symmetric functions symbolized by the specifications; and that A table may be formed expressing the expressions as linear functions of the expressions , , and the numbers occurring therein possess row and column symmetry. By solving 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 whose partition is a specification of a separation of the function symbolized by is expressible as a linear function of symmetric functions symbolized by separations of and a symmetrical table may be thus formed.” It is now to be remarked that the partition can be derived from by substituting for the numbers 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.

where is a separation of of specification placing under the summation sign to denote the specification involved,

where .

Theorem of Symmetry.—If we form the separation function

appertaining to the function each separation having a specification , multiply by and take therein the coefficient of the function we obtain the same result as if we formed the separation function in regard to the specification multiplied by and took therein the coefficient of the function

Ex. gr., take we find

The Differential Operators.—Starting with the relation

multiply each side by thus introducing a new quantity we obtain

so that a rational integral function of the elementary functions, is converted into

where

and denotes, not successive operations of but the operator of order obtained by raising to the power symbolically as in Taylor’s theorem in the Differential Calculus.

Write also so that

The introduction of the quantity converts the symmetric function into

Hence, if

Comparing coefficients of like powers of we obtain

while unless the partition contains a part Further, if denote successive operations of and

and the operations are evidently commutative.

Also and the law of operation of the operators upon a monomial symmetric function is clear.

We have obtained the equivalent operations

where denotes (by the rule over ) that the multiplication of operators is symbolic as in Taylor’s theorem. denotes, in fact, an operator of order 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

It has been shown (vide Memoir on Symmetric Functions of the Roots of Systems of Equations,” Phil. Trans. 1890, p. 490) that

where now the multiplications on the dexter denote successive operations, provided that

being an undetermined algebraic quantity.

Hence we derive the particular cases

and we can express in terms of products denoting successive operations, by the same law which expresses the elementary function in terms of the sums of powers Further, we can express in terms of by the same law which expresses the power function in terms of the elementary functions

Operation of upon a Product of Symmetric Functions.—Suppose to be a product of symmetric functions If in the identity we introduce a new root we change into and we obtain

and now expanding and equating coefficients of like powers of

the summation in a term covering every distribution of the operators of the type presenting itself in the term.

Writing these results