Estimates for the largest critical value of \(T_n^{(k)}\) (Q6593035)

For the \(n\)-th Chebyshev polynomial of the first kind \(T_n(x)=\cos n\arccos x\), \(x\in[-1,1]\), the authors study the quantity \(\tau_{n,k}:=|T_n^{(k)}(\omega)|/T_n^{(k)}(1)\), \(1\le k\le n-2\), where \(\omega=\omega_{n,k}\) is the rightmost zero of \(T_n^{(k+1)}\). This quantity measures how small is the largest critical value of \(T_n^{(k)}\) in relation to the global maximum of \(T_n^{(k)}\) over \([-1,1]\) and has a variety of applications.\N\NThe first result is the following representation: for \(n>k+1\) \[ \tau_{n,k}^2=\frac{(2k-1)!!}{(2k)!!}\frac{n}{n-k}\frac{1}{\binom{n+k}{n-k}}\frac{1}{(1-\omega^2)^k S_{n,k+1}(\omega)}, \] where \[ S_{n,k+1}(x)=1+\sum_{m=1}^k \frac{(2m-1)!!}{(2m)!!}\frac{(k+1-m)_{2m}}{(1-x^2)^m}\prod_{j=1}^m\frac{1}{n^2-j^2} \] and \((a)_j=a(a+1)\dots(a+j-1)\). This is obtained using explicit expressions of the coefficients of the Gaussian quadrature for the ultraspherical weight \((1-x^2)^{\lambda-1/2}\) with non-negative integer \(\lambda\) due to Knut Petras.\N\NUsing this new representation for \(\tau_{n,k}\) and another known one, the following estimates are derived (for \(n>k+1\)):\N\begin{align*}\N\tau_{n,k} &\le\frac{(2k-1)!!}{(n+k-1)(n+k-3)\dots(n-k+1)}\frac{1}{(1-w^2)^{\tfrac k2}}\\\N&\le\frac{(2k-1)!!}{(n+k-1)(n+k-3)\dots(n-k+1)} \left(\frac{n}{k+2}\right)^k,\N\end{align*}\Nand in the other direction \[ \tau_{n,k}\ge \frac{(2k-1)!!}{(n+k-1)(n+k-3)\dots(n-k+1)}. \]\N\NFor \(\tau_k^*=\lim_{n\to\infty}\tau_{n,k}\), it is shown that \(\tau_k^*\le \frac{(2k-1)!!}{(k+2)^k}\) and \[ \tau_k^*=A\left(\frac2e\right)^{k+1/2} e^{-ak^{1/3}} k^{-1/6}(1+O(k^{-1/6})), \] where \(A\approx 1.3951\) and \(a\approx 1.8558\), with explicit expressions for \(A\) and \(a\) given in the paper.