On three-term recurrence and Christoffel-Darboux identity for orthogonal rational functions on the real line

First, we give an algebraic proof to the Christoffel-Darboux identity of formal orthogonal rational functions on the real line by exposing some underlying algebraic properties. This proof does not involve the three-term recurrence relationship. Besides, it is shown that if a family of rational functions satisfies the Christoffel-Darboux relation, then it also admits a three-term recurrence relationship. Thus, the equivalence between both relations is revealed.