Weighted sums of orthogonal polynomials with positive zeros

We study the two sequences of polynomials which arise as denominators of the approximants of even and odd order, respectively, of a Stieltjes fraction, and which may be defined alternatively as a sequence of orthogonal polynomials with positive zeros and the associated sequence of kernel polynomials. Motivated by problems in the setting of birth-death processes, where these sequences play a major role, we focus on the asymptotic behaviour of the sequences and establish convergence of certain weighted sums of the polynomials at hand.