Inequalities for ultraspherical polynomials and application to quadrature (Q1318389)

The author derives upper bounds for the ultraspherical polynomial \(P_ n^{(\lambda)}\) in the form \((\sin \theta)^ \gamma | P_ n^{(\lambda)} (\theta) |\), \(\lambda \geq 1\), \(0 \leq \theta \leq \pi\), for various values of \(\gamma\). The results are used to give estimates for linear functionals, and in particular for the remainder term in Gaussian quadrature.