Jacobi Identities and Related Combinatorial Formulas

The derivatives of a quadratic trinomial F = ax 2 + bx + c at its roots x 1, x 2 add up to zero (Vieta’s theorem). Alternatively, we can write this as 1/F′(x 1) + 1/F′(x 2) = 0 or, equivalently, 1/(x 1 − x 2) + 1/(x 2 − x 1) = 0. Also, there is an obvious identity: x 1/(x 1 − x 2) + x 2/(x 2 − x 1) = 1. C.G.J. Jacobi’s concern with questions of dynamics and geometry led him to derive far-reaching generalizations of these identities for polynomials of arbitrary degree, which enabled him to solve several complex problems. For example, he squared an ellipsoid’s surface, determined the geodesics on this surface, and described the dynamics with two motionless centers of gravity. In this chapter, readers will become acquainted with the Jacobi identities and related formulas and their close ties with complex analysis; in addition, they will learn about some less traditional applications of the Jacobi identities (to linear differential equations). The chapter also contains notes about further generalizations and applications and a short guide to the literature.