High order (up to 4) derivatives of Chebyshev polynomials needed

Question

I need to compute high order (up to 4) derivatives of Chebyshev polynomials at the points of the Chebyshev-Lobato grid:

$$x_j=cos(πj/N), j=0,\dots,N$$

Does anyone know how to do that? I tried iterative methods but too clumsy. I also made an attempt to use the differential equation but didn't work either.

The polynomials can be defined by

$$T_n(\cos(\theta))=\cos(n\theta)$$

and I can differentiate wrt $\theta$ and use the chain rule for the first derivative but this doesn't generalize to higher derivatives. I remember seeing something like that in an old paper but now it's nowhere to be found.

Any help appreciated, Thanks.

Answer 1

Mathematica gets it via (x^2 - 1)^3*D[ChebyshevT[n, x], {x, 4}] // FullSimplify:

$ n \left(n \left(-n^2+3 (n-3)^2 x^2+4\right) U_{n-4}(x)+(n-3) x \left(3 n \left(n-2 (n-3) x^2\right) U_{n-3}(x)+(n-1) \left((n-2) x^2-3\right) U_{n-1}(x)\right)\right) $

Next you can evaluate at your desired points:

f[x_] := D[ChebyshevT[n, x], {x, 4}]
FullSimplify[f[Cos[j*\[Pi]/n]], 
 Assumptions -> {j \[Element] Integers, n \[Element] Integers}]

to get

$$ T_{n}^{(4)}(\cos(j\pi/n)) = (-1)^j n^2 \csc^4(\pi j/n) (-15 \csc^2(\pi j/n)+n^2+11) $$