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P_n(x) = x_{n-1}x_{n+1} - x_n^2, x_{n+2} = xx_{n+1} + nx_n

Source: Indian Postal Coaching 2009 set 1 p5

May 26, 2020
polynomialSequencealgebra

Problem Statement

Define a sequence <xn><x_n> by x1=1,x2=x,xn+2=xxn+1+nxn,n≥1x_1 = 1, x_2 = x, x_{n+2} = xx_{n+1} + nx_n, n \ge 1. Consider the polynomial Pn(x)=xn−1xn+1−xn2P_n(x) = x_{n-1}x_{n+1} - x_n^2, for each n≥2n \ge 2. Prove or disprove that the coefficients of Pn(x)P_n(x) are all non-negative, except for the constant term when nn is odd.
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