Editing Ring of symmetric functions (section)

=== Identities ===

The ring of symmetric functions is a convenient tool for writing identities between symmetric polynomials that are independent of the number of indeterminates: in Λ<sub>''R''</sub> there is no such number, yet by the above principle any identity in Λ<sub>''R''</sub> automatically gives identities the rings of symmetric polynomials over ''R'' in any number of indeterminates. Some fundamental identities are
:<math>\sum_{i=0}^k(-1)^ie_ih_{k-i}=0=\sum_{i=0}^k(-1)^ih_ie_{k-i}\quad\mbox{for all }k>0,</math>
which shows a symmetry between elementary and complete homogeneous symmetric functions; these relations are explained under [[complete homogeneous symmetric polynomial]].
:<math>ke_k=\sum_{i=1}^k(-1)^{i-1}p_ie_{k-i}\quad\mbox{for all }k\geq0,</math>
the [[Newton identities]], which also have a variant for complete homogeneous symmetric functions:
:<math>kh_k=\sum_{i=1}^kp_ih_{k-i}\quad\mbox{for all }k\geq0.</math>