Christoffel-Darboux type identities for independence polynomial

In this paper we introduce some Christoffel-Darboux type identities for independence polynomials. As an application, we give a new proof of a theorem of M. Chudnovsky and P. Seymour, claiming that the independence polynomial of a claw-free graph has only real roots. Another application is related to a conjecture of Merrifield and Simmons.