A sequence of polynomials , in , and is defined by and by for . Prove that each is symmetric, in other words, is unaltered by any permutation of
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A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by
$$P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)$$
for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$
Izvor: Međunarodna matematička olimpijada, shortlist 1985
Školjka 
, in 
, and 
is defined by 
and by
. Prove that each 
is symmetric, in other words, is unaltered by any permutation of 