On Estimates for the Weights in Gaussian Quadrature in the Ultraspherical Case

In this paper the Christoffel numbers av n for ultraspherical weight functions wk , wx(x) = (\ -x ) ~ ' , are investigated. Using only elementary functions, we state new inequalities, monotonicity properties and asymptotic approximations, which improve several known results. In particular, denoting by dv „ the trigonometric representation of the Gaussian nodes, we obtain for À e [0, 1] the inequalities and similar results for X <£ (0, 1). Furthermore, assuming that a\ ' remains in a fixed closed interval, lying in the interior of (0, n) as n —► oo , we show that, for every fixed A > -1/2 , awe *sin2x0w¡l A(l-A) n + * 2{n + a) sin 8:,' u ,n A(l -A)[3(A+l)(A-2) + 4sin2e^J 8(« + /.)4sin4Ö v , n 0(n 7)