KöMaL 2018 B4997

Consider the following sequence $p_n(x)$ of polynomials with integer coefficients: let $p_0(x)=0$, $p_1(x)=1$, and for all $n\ge 2$ let $p_n(x)=p_{n-1}(x)+x\cdot p_{n-2}(x).$ Prove that if a polynomial $f(x)$ divides the polynomials $p_n(x)$ and $p_m(x)$ for some positive integers $n$, $m$ then it also divides the polynomial $p_{(m,n)}(x)$. ($(n,m)$ denotes the greatest common divisor of $n$ and $m$. A polynomial $P(x)$ divides a polynomial $Q(x)$ if there exists a polynomial $R(x)$ of real coefficients for which $Q(x)=P(x)R(x)$.)