THE COMPLEMENTARY SYMMETRICAL FUNCTIONS - CONNECTION CONSTANTS USING NEGATIVE SETS

The coefficients relating the powers of x with sequences of the form p (0)(x) = 1, p(n)(x) = (x - a(n)) p(n-1)(x), for any n > 0. are well kn own: in one sense, we have the elementary symmetric functions, and in the other, the complete symmetric functions. In this paper, through th e use of the theory of negative sets, we give expansions of rational f unctions in terms of sequences of the above form extended as follows f or n < 0: p(n)(x) = (x - a(n))(-1) p(n+1)(x). As an application, we ge neralize many well-known combinatorial identities. (C) 1998 Academic P ress.