A sequence of polynomials
![P_m(x, y, z), m = 0, 1, 2, \cdots](/media/m/b/9/7/b977197c6a565e7f48855c38d1ae4410.png)
, in
![x, y](/media/m/2/7/9/279a699b10f7b70e7160f4aaaa89e453.png)
, and
![z](/media/m/d/2/4/d241a79f1fdd0ce9a8f3f91570ba5d62.png)
is defined by
![P_0(x, y, z) = 1](/media/m/8/1/2/812f199b365a4bdd8995f64cc72b33ca.png)
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)](/media/m/e/c/2/ec23ae4878d50906ff094624aa9689ab.png)
for
![m > 0](/media/m/3/8/a/38a72e72d0dbf59ffee8888ee12a2bc8.png)
. Prove that each
![P_m(x, y, z)](/media/m/5/9/d/59da942b7786af0191d623c0bfd1f47d.png)
is symmetric, in other words, is unaltered by any permutation of
%V0
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 $x, y, z.$