Christoffel-Darboux-type formulae for orthonormal rational functions with arbitrary complex poles

Consider the reproducing kernel K-n(x, y) = Sigma(n-1)(k= 0) phi k (x) <(phi k (y)) over bar> where {phi k}(k=0)(n-1) forms an orthonormal basis for the space of rational functions with poles among {alpha(1), alpha(2), ..., alpha(n-1)} subset of C. In this paper, we then derive Christoffel-Darboux-type formulae for K-n by exploiting its relation with so-called quasi-orthogonal rational functions Q(n) of the form Q(n)(x)= a(n)phi(n)(x) + b(n)((1 -x/(alpha) over bar (n)-1)/(1 - x/alpha(n)))phi(n-1) (x), where a(n), b(n) epsilon C. We conclude with an application in rational Gaussian quadrature and in rational Krylov methods.