The Gauss–Jacobi quadrature rule

Assume Countable Choice, let $n\ge 1$, and let $\alpha ,\beta >-1$. Let ${\mu }_{\alpha ,\beta }$ be the finite positive Borel measure with density $(1-t{)}^{\alpha }(1+t{)}^{\beta }$ on $(-1,1)$ from The Jacobi weight defines a finite positive measure with full support. Its moments through degree $2n-1$ are finite and its support is all of $[-1,1]$, so it is Gaussian moment data of order $2n-1$ in Positive-measure Gaussian quadrature moment data. The $n$-node Gauss–Jacobi rule is the general Gaussian rule $Q_n$ for this measure as defined in The m-node Gaussian quadrature rule for a positive measure: its nodes are the roots of the monic degree-$n$ orthogonal polynomial, and each weight is the integral of the corresponding Lagrange cardinal polynomial. This definition makes no accuracy or computed-node claim; local well-definedness is supplied by Gauss–Jacobi nodes and weights are well-defined ↗.