Is there a formula expressing the top coeffient of the Lagrange interpolation polynomial for a function as an average of its ($n-1$)-st derivative (divided by $(n-1)!$)?
I am looking for a reference; such a formula does not appear in standard references I saw.
In notation. Let $f:\mathbb{C}\to \mathbb{C}$ be a smooth enough function. Let $L_{f;x_1,\dotsc,x_n}(x)=a_{n-1}x^{n-1}+\dotsb+a_0$ be the Lagrange interpolation polynomial for $f$ at points $x_1,\dotsc,x_n$. I am looking for formulas expressing $a_{n-1}$ in terms of $f^{(n-1)}$, or in fact as its average $$a_{n-1}=\frac{1}{(n-1)!} \int_x f^{(n-1)}(x)d\mu_{(x_1,\dotsc,x_n)}$$ for some measure $\mu$ depending on $x_1,\dotsc,x_n$.
In the degenerate case $x_1=\dotsb=x_n$ this is true: the Lagrange interpolation polynomial of $f$ is its Taylor polynomial $\sum_{i=0}^{i=n-1} \frac {f^{(i)}(x_1)}{i!} x^i$, and $\mu$ is the delta measure at $x_1=\dotsb=x_n$.
