Page:An introduction to Combinatory analysis (Percy MacMahon, 1920, IA Introductiontoco00macmrich).djvu/19

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Elementary theory of Symmetric Functions

5

If we add all such functions which have the same weight we obtain, algebraically speaking, all the products together of the quantities , repetitions permissible.

Such a sum is called the Homogeneous Product-Sum of weight of the quantities.

It is usually denoted by .

We have

and so forth.

We have before us the three sets of functions

The first and third sets contain an infinite number of members, but the second set only involves members where is the number of the quantities .


6. The identity of Art. 4 which connects the functions with may be written, by putting for , , or in the form . If we expand the last fraction in ascending powers of , we obtain, in the first place, .
It is clear that the coefficient of is the homogeneous product-sum of weight , so that we may write an identity.