A direct proof of the Christoffel-Darboux identity and its equivalence to the recurrence relationship

摘要

First a direct proof of the Christoffel-Darboux identity for orthogonal polynomials is given. This proof does not make use of the three-terms recurrence relationship. Then it is proved that if a family of polynomials satisfies the Christoffel-Darboux identity, then it also satisfies a three-terms recurrence relationship and thus it forms a family of orthogonal polynomials which shows the equivalence between both relations.