I'm currently dealing with the following kind of two-term recursion for real polynomials $p_{k}(x)$, $k \in \mathbb{N}$: \begin{eqnarray} p_{0}(x) &=& 1, \\ p_{1}(x) &=& 1-2x, \\ p_{k+1}(x) &=& -p_{k-1}(x) + p_{k}(x)\left(2 - 4x\right). \end{eqnarray}
Rewinding the recursion for the first few polynomials would give the following expanded forms: \begin{eqnarray} p_{2}(x) &=& 1 - 8 x + 8 x^2, \\ p_{3}(x) &=& 1 - 18 x + 48 x^2 - 32 x^3, \\ p_{4}(x) &=& 1 - 32 x + 160 x^2 - 256 x^3 + 128 x^4, \\ p_{5}(x) &=& 1 - 50 x + 400 x^2 - 1120 x^3 + 1280 x^4 - 512 x^5 \\ p_{6}(x) &=& 1 - 72 x + 840 x^2 - 3584 x^3 + 6912 x^4 - 6144 x^5 + 2048 x^6. \end{eqnarray} Graph of the first four polynomials.
The above plots lead to the conclusion that all roots are within the interval $(0,1)$ and that $|p_{k}(x)| \leq 1$, if $0 \leq x \leq 1$.
Does anyone know a good way to actually verify/proof the above hypothesis for this kind of two-term recursion?