On the parity of the first kind Chebyshev polynomial over the positive integers

Question

Let us consider the first kind Chebyshev polynomial over the positive integers $T_{n+1}(x) = 2xT_n(x) - T_{n-1}(x)$ with $n>2$ is an odd number.

We have

$$T_0(x) = 1$$

$$T_1(x) = x $$

$$T_3(x) = 4x^3 − 3x $$

My question is:

(1) How one can determine when $T_n(x)$ have even values and

(2) How one can determine when $T_n(x)$ have odd values

So, the problem is the study of the parity of the function $T_n(x)$ over the positive integers.

Answer 1

As you already note, the Chebyshev polynomials of the first kind satisfy the recurrence $$T_{n+1}(x)=2xT_n(x)-T_{n-1}(x),$$ for every integer $n\geq1$, with $T_0=1$ and $T_1=x$. It follows that for every integer $n\geq1$ you have $$T_{n+1}(x)\equiv T_{n-1}(x)\pmod{2},$$ and hence that for every integer $x$ \begin{eqnarray*} T_n(x)\equiv T_0(x)=1\pmod{2}\qquad\text{ if $x$ is even}\\ T_n(x)\equiv T_1(x)=x\pmod{2}\qquad\text{ if $x$ is odd}\hspace{3pt} \end{eqnarray*} So $T_n(x)$ is odd if and only if $n$ and $x$ are odd.