Lagrange Polynomial Interpolation, centered coefficients

Question

My textbook states that the Lagrange Interpolant on the interval $[a,b]$, with the data points $(x_0,y_0),...,(x_n,y_n)$, written as:
$\prod_nf(x)=\sum_{i=0}^ny_i\phi_i(x)$, with $\phi_i(x)$ being the Lagrange polynomials, can also be written $$\prod_nf(x)=\sum_{i=0}^nc_i(x-m)^{n-i}$$ Where $m=(a+b)/2$. I am not sure how to compute to coefficients $c_i$. How is this done? I simply computed the coefficients for $n=1,2$, but failed to establish a pattern.

Answer 1

If I understood your question correctly, two answers linked below should help solve your problem

1. if $p(x)=a_n x^n + a_{n-1}x^{n-1}+\cdots+a_0$ then $\displaystyle a_n = \sum_{i=0}^n y_i \prod_{j=0,j\neq i}^n \frac{1}{x_i-x_j}$
2. recentering a polynomial: $ \sum_{k=0}^p a_k \left(z+z'\right)^k = \sum_{\ell=0}^p \left(\sum_{k=l}^p \binom{k}{\ell} (z')^{k-\ell} a_k \right) z^\ell=\sum_{\ell=0}^p b_{\ell}\, z^\ell $