Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: q-alg
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The categories of this paper are: physics
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an
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The categories of this paper are: stat
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The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ph
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The categories of this paper are: q-fin