E. Damiani et O. Dantona, THE COMPLEMENTARY SYMMETRICAL FUNCTIONS - CONNECTION CONSTANTS USING NEGATIVE SETS, Advances in mathematics, 135(2), 1998, pp. 207-219
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.