Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/20

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6

Elementary theory of Symmetric Functions

Thence we obtain .

Since this is an identity we may multiply out the left-hand side and equate the coefficients of the successive powers of to zero; obtaining relations which enable us to express any function in terms of members of the series .


7. In the applications to combinatory analysis it usually happens that we may regard as being indefinitely great and then the relations are simply continued indefinitely.

The before-written identity now becomes , and herein writing for and transposing the factors we find , an identity which is derivable from the former by interchange of the symbols and .

There is thus perfect symmetry between the symbols and it follows as a matter of course that in any relation connecting the quantities with the quantities we are at liberty to interchange the symbols , . This interesting fact can be at once verified in the case of the relations , etc.

Solving these equations we find