**Background and notations:** Given an interval $I\subseteq \mathbb{R}$ and a continuous finite measure $d\mu = w(x)dx$, and denote $p_n(x)$ the orthogonal polynomials with respect to $d\mu$. We have the following recurrence relation $$ p_j (x) = (a_j x +b_j)p_{j-1}(x) +c_jp_{j-2}(x) \, ,\quad \forall j\geq 1\, \quad p_0(x) \equiv 1,\,p_{-1}(x)\equiv 0 \, ,$$ where the constants are determined by the measure.

In the famous Golub-Welsch paper, section 4, they give a numerical method to calculate the constants. However, it requires the numerical value of $\int\limits_{I}x^\ell w(x)\,dx$ for all non-negative integers $\ell$.

**The problem** is that for numerical integration we usually need a quadrature formula, for which weights we need the recurrence relation (see the other sections of the same paper, for example).

**Question 1:** Is there a way to compute $\int\limits_{I}x^\ell w(x)\,dx$ without any quadrature formula?

**Question 2:** Is there a way to compute the recurrence constants without evaluating integrals?

*Remark:* We can always evaluate $\int\limits_{I}x^\ell w(x)\,dx$ using the Gauss Legendre quadrature. This means that we need only the well-known recurrence for the Lebesgue measure to compute these integrals for all other continuous measures. I'm looking for something else, though.