On some problems of P. Turán concerning \(L_m\) extremal polynomials and quadrature formulas (Q1125455)

Let be \(w(x)\) a weight function on \([-1,1]\). It is known that for each \(n\in\mathbb{N}\) there exists a unique extremal polynomial \(P_n(w,m;x)= x^n+\dots\) for which \[ \int^1_{-1}P_n(w,m;x)^m w(x)dx= \min_{P=x^n+ \cdots} \int^1_{-1} P(x)^mw(x)dx. \] The zeros of the polynomial \(P_n(w,m;x)\) are real, distinct and located in the interior of the interval \([-1,1]\). This paper deals with the explicit form of the extremal polynomials with respect to the weights \((1-x)^{-1/2} (1+x)^{m-1/2}\) and \((1-x)^{m-1/2} (1+x)^{-1/2}\) for even \(m\). The author also gives an explicit representation for the Cotes numbers of the corresponding Turán quadrature formulas and their asymptotic behavior.