Problem

Source: Iran 3rd round 2014 - final exam problem 8

Tags: algebra, polynomial, algebra unsolved



The polynomials $k_n(x_1, \ldots, x_n)$, where $n$ is a non-negative integer, satisfy the following conditions \[k_0=1\] \[k_1(x_1)=x_1\] \[k_n(x_1, \ldots, x_n) = x_nk_{n-1}(x_1, \ldots , x_{n-1}) + (x_n^2+x_{n-1}^2)k_{n-2}(x_1,\ldots,x_{n-2})\] Prove that for each non-negative $n$ we have $k_n(x_1,\ldots,x_n)=k_n(x_n,\ldots,x_1)$.