Question 36
Define a sequence $\langle f_{0}(x), f_{1}(x), f_{2}(x), \ldots \rangle$ of functions by
$$
f_{0}(x)=1, \quad f_{1}(x)=x, \quad (f_{n}(x))^{2}-1=f_{n+1}(x) f_{n-1}(x), \text{ for } n \geq 1
$$
Prove that each $f_{n}(x)$ is a polynomial with integer coefficients.
$$
f_{0}(x)=1, \quad f_{1}(x)=x, \quad (f_{n}(x))^{2}-1=f_{n+1}(x) f_{n-1}(x), \text{ for } n \geq 1
$$
Prove that each $f_{n}(x)$ is a polynomial with integer coefficients.