Chebyshev polynomials of the first and second kinds by their three-term recurrences

Let $\mathcal P_{\mathbb R}$ be the set of formal real polynomials of Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials. Associate to every $P\in\mathcal P_{\mathbb R}$ an eventually zero coefficient sequence $(p_k)$: use its displayed finite coefficient list and extend it by zeros when $P\ne0$, and put $p_k=0$ for every $k$ when $P=0$. Do the same for $Q$, with coefficients $(q_k)$. Define $P+Q$ coefficientwise and define $PQ$ by the convolution coefficients

$$(P+Q)_k=p_k+q_k,\qquad (PQ)_k=\sum_{j=0}^{k}p_jq_{k-j}.$$

using the real finite sum of Finite sums and finite products, by recursion. In either construction, discard trailing zero coefficients; if every coefficient is zero, the result is the zero polynomial. Scalar multiplication and subtraction are the corresponding coefficientwise operations. Thus these formulas define operations on the formal coefficient-list objects, independently of evaluation.

Write $1=(1)$ and $X=(0,1)$, and define

$$\Phi(P,Q):=(Q,2XQ-P)\qquad(P,Q\in\mathcal P_{\mathbb R}).$$

Apply The recursion theorem to the set $\mathcal P_{\mathbb R}\times\mathcal P_{\mathbb R}$, first with initial value $(1,X)$ and then with initial value $(1,2X)$, always using the function $\Phi$. This gives unique pair sequences $H^T,H^U$. Define $T_n$ and $U_n$ to be the first coordinates of $H^T_n$ and $H^U_n$, respectively. Since the first coordinate of $\Phi(P,Q)$ is $Q$, the second coordinate of $H^T_n$ is $T_{n+1}$ and the second coordinate of $H^U_n$ is $U_{n+1}$. Consequently

$$T_0=1,\quad T_1=X,\quad T_{n+2}=2XT_{n+1}-T_n,$$ $$U_0=1,\quad U_1=2X,\quad U_{n+2}=2XU_{n+1}-U_n$$

for every $n\in\mathbb N$. These unique sequences are the Chebyshev polynomials of the first and second kinds.