I am looking for simple formulas for higher-order derivatives of Legendre polynomials $P^{(k)}_{n}(\pm 1)$. For the Chebyshev polynomials, there is a simple formula
$$ \left. \frac{d^{p}T_n}{dx^p}\right|_{x = \pm 1} = (\pm 1)^{n+p} \prod_{k=0}^{p-1} \frac{n^2 -k^2}{2k+1} $$
More generally, I am interested in the following question: Are there simple formulas for higher derivatives at endpoints $a$ and $b$ for any function orthogonal wrt a weight function $w$ on $[a,b]$? For example, I'm interested in the same question for Gegenbaurer and Jacobi polynomials as well, but partial results are useful.
Consulting Chapter 22 of Abramowitz and Stegun
$$ \frac{d^m}{dx^m}P_n(x)=1\cdot3\ldots(2m-1)C^{(m+1/2)}_{n-m}(x)\\ C^{(\alpha)}_{n}(1)=\binom{n+2\alpha-1}{n}\\ C^{(\alpha)}_{n}(-x)=(-1)^nC^{(\alpha)}_{n}(x) $$