Let $I=(a,b)\subset \mathbb{R}$ be an interval, let $w:I\to[0,\infty)$ be a measurable weight with $w>0$ $\mathcal{L}^1$-a.e. on $I$, and assume all polynomial moments against $w\,d\mathcal{L}^1$ appearing below are finite. Let $(p_j)_{j=0}^{\infty}$ be a sequence of real orthogonal polynomials, where each $p_j\in \mathbb{R}[t]$ has degree $j$ and leading coefficient $k_j\neq 0$, and where

\begin{align*} \int_I p_i(t)p_j(t)w(t)\,d\mathcal{L}^1(t)=0 \end{align*}

whenever $i\neq j$. Define the squared norm $h_j>0$ by

\begin{align*} h_j=\int_I p_j(t)^2w(t)\,d\mathcal{L}^1(t). \end{align*}

Then, for every $n\in\mathbb{N}\cup\{0\}$ and every $x,y\in\mathbb{R}$ with $x\neq y$,

\begin{align*} \sum_{j=0}^{n}\frac{p_j(x)p_j(y)}{h_j}=\frac{k_n}{k_{n+1}h_n}\frac{p_{n+1}(x)p_n(y)-p_n(x)p_{n+1}(y)}{x-y}. \end{align*}