Question 12
3b. (GBR 4) A sequence of polynomials $P_m(x,y,z)$, $m=0,1,2,\ldots$, 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 $x,y,z$.
$$P_m(x,y,z)=(x+z)(y+z)P_{m-1}(x,y,z+1)-z^2P_{m-1}(x,y,z)$$
for $m>0$. Prove that each $P_m(x,y,z)$ is symmetric, in other words, is unaltered by any permutation of $x,y,z$.