Generating Functions
The first definition of ΛR as a subring of R] allows expression the generating functions of several sequences of symmetric functions to be elegantly expressed. Contrary to the relations mentioned earlier, which are internal to ΛR, these expressions involve operations taking place in R] but outside its subring ΛR], so they are meaningful only if symmetric functions are viewed as formal power series in indeterminates Xi. We shall write "(X)" after the symmetric functions to stress this interpretation.
The generating function for the elementary symmetric functions is
Similarly one has for complete homogeneous symmetric functions
The obvious fact that explains the symmetry between elementary and complete homogeneous symmetric functions. The generating function for the power sum symmetric functions can be expressed as
((Macdonald, 1979) defines P(t) as Σk>0 pk(X)tk−1, and its expressions therefore lack a factor t with respect to those given here). The two final expressions, involving the formal derivatives of the generating functions E(t) and H(t), imply Newton's identities and their variants for the complete homogeneous symmetric functions. These expressions are sometimes written as
which amounts to the same, but requires that R contain the rational numbers, so that the logarithm of power series with constant term 1 is defined (by ).
Read more about this topic: Ring Of Symmetric Functions, Properties of The Ring of Symmetric Functions
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