Problem A. 379. (September 2005)
A. 379. Find all real numbers for which there exists a nonzero polynomial P, such that
for all n. List all such polynomials for =2.
(5 pont)
Deadline expired on 17 October 2005.
Sketch of solution. Let P(x)=a_{0}+a_{1}x+...+a_{k}x^{k}. Both sides of the equation contains a polynomial of n, degree is k. The coefficients must pairwise match.
Since the coefficient of n^{k} is on the lefthand side, and it is a_{k} on the righthand side. Therefore, .
If then we have a homogenous system of linear equations for the coefficients and one of these equations vanishes. So the system has infinitely many solutions.
For =2 we obtain k=3 and P(x)=c(x^{3}x).
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