# Top coefficient of the Lagrange polynomial as average of (n-1)-st derivative

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 $$x_1=\dotsb=x_n$$.