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Polynomial and equation

Source: Austrain Polish 1996

December 7, 2013
algebrapolynomialrecurrence relation

Problem Statement

The polynomials Pn(x)P_{n}(x) are defined by P0(x)=0,P1(x)=xP_{0}(x)=0,P_{1}(x)=x and P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x)   n\geq 2 For every natural number n≥1n\geq 1, find all real numbers xx satisfying the equation Pn(x)=0P_{n}(x)=0.
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