Minimizing the error function of Gauss-Jacobi quadrature rule with respect to parameters \(\alpha\) and \(\beta\) (Q2493749)

The error unction of Gauss-Jacobi quadrature rule is \[ Z_{n}(\alpha, \beta)= \frac{f^{(2n)}(\xi)}{(2n)!} \cdot \frac{2^{\alpha+ \beta+2n+1}n!\Gamma (\alpha+n+1) \Gamma (\beta+n+1) \Gamma(\alpha+ \beta+n+1) }{(\alpha+\beta+2n-1)\Gamma (\alpha+\beta+2n+1)^{2} }, \] \[ \alpha, \beta \geq -1; -1 \leq \xi \leq 1 \] If \(| f^{(n)}(x)| <M, x\in [-1, 1], M\)-a constant, then the authors optimize this function for the values of \(\alpha\) and \(\beta\). Some numerical examples are given