Suppose there is a finite sequence $(a_0, a_1, a_2, \cdots, a_k)$, is there any way to use a polynomial $\alpha_2 x^2 + \alpha_1 x + \alpha_0$ to generate that sequence such that each coefficient could be given with one set of $\alpha$, i.e. $a_j=\alpha_2 j^2 + \alpha_1 j + \alpha_0$?

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    $\begingroup$ The notation $\Sigma_m$ is rather vague. Please be as precise as possible. $\endgroup$ – Cameron Buie Oct 10 '13 at 5:03
  • $\begingroup$ I updated the question and let's consider the simple case with only degree-2 polynomial first. Thanks. $\endgroup$ – user1285419 Oct 10 '13 at 5:07

To prescribe $k+1$ values of a polynomial, you need in general a polynomial of degree $k$, though you might get lucky and have the values turn out to fit a lower-degree polynomial perfectly. See http://en.wikipedia.org/wiki/Lagrange_polynomial


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