Explicit orthogonal polynomials for reciprocal polynomial weights on $(-\infty ,\infty )$

Let $S$ be a polynomial of degree $2n+2$, that is, positive on the real axis, and let $w=1/S$ on $(-\infty ,\infty )$. We present an explicit formula for the $n$th orthogonal polynomial and related quantities for the weight $w$. This is an analogue for the real line of the classical Bernstein-Szegő formula for $\left(-1,1\right)$.