Ring of symmetric functions: Difference between revisions

Revision as of 07:06, 15 August 2018

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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;_''k''>0&nbsp;''p''_''k''(''X'')''t''^''k''−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

:<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>).