On the convergence of closed interpolatory integration rules based on the zeros of Gegenbauer polynomials (Q579560)

The Gegenbauer polynomials \(C_ n^{\mu}(x)\) are orthogonal with respect to the weight function \((1-x^ 2)^{\mu -}\). Consider the interpolatory integration rules \(Q_ nf=\sum^{n}_{i=1}w_{in}f(x_{in}),\) where \(x_{in}\) are the zeros of \((1-x^ 2)C^{\mu}_{n-2}(x)\) and \[ w_{in}=\int^{1}_{- 1}\frac{(1-x^ 2)C^{\mu}_{n-2}(x)dx}{(x-x_{in})[(1-x^ 2)C^{\mu}_{n-2}(x)]'_{x=x_{in}}}. \] It is known [see \textit{G. Pólya}, Math. Z. 37, 264-286 (1933; Zbl 0007.00703)] that if \(\leq \mu \leq 4\), then the sequence \(\{Q_ nf\}\) converges to \(If=\int^{1}_{- 1}f(x)dx\) for all Riemann-integrable functions on \([-1,1].\) The author proves that this holds also for \(-<\mu <\).