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Sequence of polynomials

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August 29, 2010
algebrapolynomialfunctional equationrecurrence relationsymmetryIMO Shortlist

Problem Statement

A sequence of polynomials Pm(x,y,z),m=0,1,2,⋯P_m(x, y, z), m = 0, 1, 2, \cdots, in x,yx, y, and zz is defined by P0(x,y,z)=1P_0(x, y, z) = 1 and by Pm(x,y,z)=(x+z)(y+z)Pm−1(x,y,z+1)−z2Pm−1(x,y,z)P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z) for m>0m > 0. Prove that each Pm(x,y,z)P_m(x, y, z) is symmetric, in other words, is unaltered by any permutation of x,y,z.x, y, z.
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