The first definition of Λ<sub>''R''</sub> as a subring of ''R''[<span/>[''X''<sub>1</sub>,''X''<sub>2</sub>,…]] allows expression 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''[<span/>[''X''<sub>1</sub>,''X''<sub>2</sub>,…;''t'']] but outside its subring Λ<sub>''R''</sub>[<span/>[''t'']], 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
