Coefficients of Lagrange polynomials Let $n\in\mathbb{N}^*,A=(a_1,\dots,a_n)\in\mathbb{K}[X]^n$ all different numbers and $B=(b_1,...,b_n)\in\mathbb{K}[X]^n$ all different numbers.
Let $L_{A,B}$ be the polynomial of degree $n-1$ verifying $\forall i\in[|1,n|],L_{A,B}(a_i)=b_i$. ($[|1,n|]=\{1,2,\dots,n\}$)
We know that this is a Lagrange interpolation polynomial and can be written $\displaystyle L_{A,B}(X)=\sum_{i=1}^n b_i\prod_{k=1,k\neq i}^n\dfrac{X-a_k}{a_i-a_k}$
However, that gives us a pretty 'abstract' definition of the polynomial. What is a good formula of the coefficient $C_k$ before $X^k$ in $L_{A,B}(X)$ ?
 A: You can get a closed-form expression for Lagrange coefficients if you use a different representation. "Beginner's guide to mapping simplexes affinely", section "Lagrange interpolation", describes a determinant form of Lagrange polynomial that interpolates $(a_0;b_0)$, $\dots$, $(a_n;b_n)$
$$
P(x) = (-1)
\frac{
    \det
    \begin{pmatrix}
        0       & b_0       & b_1       & \cdots & b_n       \\
        x^n     & a_0^n     & a_1^n     & \cdots & a_n^n     \\
        x^{n-1} & a_0^{n-1} & a_1^{n-1} & \cdots & a_n^{n-1} \\
        \cdots  & \cdots    & \cdots    & \cdots & \cdots    \\
        1       & 1         & 1         & \cdots & 1         \\
    \end{pmatrix}
}{
    \det
    \begin{pmatrix}
        a_0^n     & a_1^n     & \cdots & a_n^n     \\
        a_0^{n-1} & a_1^{n-1} & \cdots & a_n^{n-1} \\
        \cdots    & \cdots    & \cdots & \cdots    \\
        1         & 1         & \cdots & 1         \\
    \end{pmatrix}
}.
$$
Using Laplace expansion along the first column in the numerator you can get expressions for coefficients at $x^i$.
Result should look as follows
$$
c_i = (-1)^{n-i}
\frac{
    \det
    \begin{pmatrix}
        b_0       & b_1       & \cdots & b_n       \\
        a_0^n     & a_1^n     & \cdots & a_n^n     \\
%        x_0^{n-1} & x_1^{n-1} & \cdots & x_n^{n-1} \\
        \cdots    & \cdots    & \cdots & \cdots    \\
        a_0^{i-1} & a_1^{i-1} & \cdots & a_n^{i-1} \\
        a_0^{i+1} & a_1^{i+1} & \cdots & a_n^{i+1} \\
        \cdots    & \cdots    & \cdots & \cdots    \\
        1         & 1         & \cdots & 1         \\
    \end{pmatrix}
}{
    \det
    \begin{pmatrix}
        a_0^n     & a_1^n     & \cdots & a_n^n     \\
        a_0^{n-1} & a_1^{n-1} & \cdots & a_n^{n-1} \\
        \cdots    & \cdots    & \cdots & \cdots    \\
        1         & 1         & \cdots & 1         \\
    \end{pmatrix}
},
$$
where $c_i$ is coefficient at $x^i$ in the polynomial.
For practical example you may want to check "Workbook on mapping simplexes affinely", section "Lagrange interpolation".
A: Take a polynome P of degree n, it could be writen: 
$P(x) = c_nx^n + c_{n-1}x^{n-1} + \cdots + c_0 $
or 
$ P(x) = c_n(x-r_1)(x-r_2)\cdots(x-r_n)\ $
You can then define symetrical polynome:
$\sigma_1(r_1,...,r_n)=\sum_{i=1}^n r_i = r_1 + \cdots + r_n$
$\sigma_2(r_1,...,r_n)=\sum_{1\le i<j\le n} r_ir_j = r_1 r_2 + \cdots + r_{n-1} r_n $
$\sigma_k(r_1,...,r_n)=\sum_{1\le i_1<\cdots<i_k\le n} r_{i_1}r_{i_2}\ldots r_{i_k} $
$\sigma_n(r_1,...,r_n)=r_1r_2\ldots r_n$
Or in other words, $\sigma_k$ is the sum of products of k roots.
Then you have the relatonship:
$\sigma_{k}=(-1)^{k}\cdot\frac{c_{n-k}}{c_{n}}$
Keeping in mind that the lagrange polynomes are of degree n-1, the coefficient for $X^k$ in $L_i = b_i \prod_{k=1,k\neq i}^n\dfrac{X-a_k}{a_i-a_k}$ is given by:
$c_{n-1} = b_i \prod_{k=1,k\neq i}^n\dfrac{1}{a_i-a_k}$
$ c_k = c_{n-1} (-1)^{n-1-k} \sigma_{n-1-k}(a_1,..., a_{i-1},a_{i+1},...,a_n)$ $ = b_i (\prod_{k=1,k\neq i}^n\dfrac{1}{a_i-a_k})(-1)^{n-1-k} \sigma_{n-1-k} (a_1,..., a_{i-1},a_{i+1},...,a_n)$
Then you can sum for i: 
$c_{n-1} = \sum_{i=1}^n b_i \prod_{k=1,k\neq i}^n\dfrac{1}{a_i-a_k}$
$ c_k = \sum_{i=1}^n b_i (\prod_{k=1,k\neq i}^n\dfrac{1}{a_i-a_k})(-1)^{n-1-k} \sigma_{n-1-k} (a_1,..., a_{i-1},a_{i+1},...,a_n)$
Note: I can't think of a situation where this would be handy.
