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

Here we study the quantity $$\tau_{n,k}:=\frac{|T_n^{(k)}(\omega_{n,k})|}{T_n^{(k)}(1)}\,,$$ where $T_n$ is the $n$-th Chebyshev polynomial of the first kind and $\omega_{n,k}$ is the largest zero of $T_n^{(k+1)}$. Since the absolute values of the local extrema of $T_n^{(k)}$ increase monotonically towards the end-points of $[-1,1]$, the value $\tau_{n,k}$ shows how small is the largest critical value of $\,T_n^{(k)}\,$ relative to its global maximum $\,T_n^{(k)}(1)$. This is a continuation of the recent paper \cite{NNS2018}, where upper bounds and asymptotic formuae for $\tau_{n,k}$ have been obtained on the basis of Alexei Shadrin's explicit form of the Schaeffer--Duffin pointwise majorant for polynomials with absolute value not exceeding $1$ in $[-1,1]$. We exploit a result of Knut Petras \cite{KP1996} about the weights of the Gaussian quadrature formulae associated with the ultraspherical weight function $w_{\lambda}(x)=(1-x^2)^{\lambda-1/2}$ to find an explicit (modulo $\omega_{n,k}$) formula for $\tau_{n,k}^2$. This enables us to prove a lower bound and to refine the upper bounds for $\tau_{n,k}$ obtained in \cite{NNS2018}. The explicit formula admits also a new derivation of the assymptotic formula in \cite{NNS2018} approximating $\tau_{n,k}$ for $n\to\infty$. The new approach is simpler, without using deep results about the ordinates of the Bessel function, and allows to better analyze the sharpness of the estimates.