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3

Part of 1972 Czech and Slovak Olympiad III A

Problems(1)

Sequence of polynomials with recursion

Source: Czech and Slovak Olympiad 1972, National Round, Problem 3

7/10/2024
Consider a sequence of polynomials such that P0(x)=2,P1(x)=xP_0(x)=2,P_1(x)=x and for all n≥1n\ge1 Pn+1(x)+Pn−1(x)=xPn(x).P_{n+1}(x)+P_{n-1}(x)=xP_n(x). a) Determine the polynomial Qn(x)=Pn2(x)−xPn(x)Pn−1(x)+Pn−12(x)Q_n(x)=P^2_n(x)-xP_n(x)P_{n-1}(x)+P^2_{n-1}(x) for n=1972.n=1972. b) Express the polynomial (Pn+1(x)−Pn−1(x))2\bigl(P_{n+1}(x)-P_{n-1}(x)\bigr)^2 in terms of Pn(x),Qn(x).P_n(x),Q_n(x).
algebrapolynomialSequencerecurrence relation