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Sequence of polynomials - ISL 1971

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September 22, 2010
algebrapolynomialalgebraic identitiesrecurrence relationIMO Shortlist

Problem Statement

Consider a sequence of polynomials P0(x),P1(x),P2(x),…,Pn(x),…P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots, where P0(x)=2,P1(x)=xP_0(x) = 2, P_1(x) = x and for every n≥1n \geq 1 the following equality holds: Pn+1(x)+Pn−1(x)=xPn(x).P_{n+1}(x) + P_{n-1}(x) = xP_n(x). Prove that there exist three real numbers a,b,ca, b, c such that for all n≥1,n \geq 1, (x2−4)[Pn2(x)−4]=[aPn+1(x)+bPn(x)+cPn−1(x)]2.(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.
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