Editing Ring of symmetric functions (section)

=== Generating functions ===

The first definition of Λ<sub>''R''</sub> as a subring of <math>R[[X_1, X_2, ...]]</math> allows the [[generating function]]s of several sequences of symmetric functions to be elegantly expressed. Contrary to the relations mentioned earlier, which are internal to Λ<sub>''R''</sub>, these expressions involve operations taking place in ''R''<nowiki>[[</nowiki>''X''<sub>1</sub>,''X''<sub>2</sub>,...;''t''<nowiki>]]</nowiki> but outside its subring Λ<sub>''R''</sub><nowiki>[[</nowiki>''t''<nowiki>]]</nowiki>, so they are meaningful only if symmetric functions are viewed as formal power series in indeterminates ''X''<sub>''i''</sub>. We shall write "(''X'')" after the symmetric functions to stress this interpretation.

The generating function for the elementary symmetric functions is
:<math>E(t) = \sum_{k \geq 0} e_k(X)t^k = \prod_{i=1}^\infty (1+X_it).</math>
Similarly one has for complete homogeneous symmetric functions
:<math>H(t) = \sum_{k \geq 0} h_k(X)t^k = \prod_{i=1}^\infty \left(\sum_{k \geq 0} (X_it)^k\right) = \prod_{i=1}^\infty \frac1{1-X_it}.</math>
The obvious fact that <math>E(-t)H(t) = 1 = E(t)H(-t)</math> explains the symmetry between elementary and complete homogeneous symmetric functions.
The generating function for the power sum symmetric functions can be expressed as
:<math>P(t) = \sum_{k>0} p_k(X)t^k = \sum_{k>0}\sum_{i=1}^\infty (X_it)^k = \sum_{i=1}^\infty\frac{X_it}{1-X_it} = \frac{tE'(-t)}{E(-t)} = \frac{tH'(t)}{H(t)}</math>
((Macdonald, 1979) defines ''P''(''t'') as Σ<sub>''k''>0</sub>&nbsp;''p''<sub>''k''</sub>(''X'')''t''<sup>''k''−1</sup>, and its expressions therefore lack a factor ''t'' with respect to those given here). The two final expressions, involving the [[formal derivative]]s 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
:<math>P(t) = -t\frac d{dt}\log(E(-t)) = t\frac d{dt}\log(H(t)),</math>
which amounts to the same, but requires that ''R'' contain the rational numbers, so that the logarithm of power series with constant term&nbsp;1 is defined (by <math>\textstyle\log(1-tS) = -\sum_{i>0} \frac1i(tS)^i</math>).