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 $P\mathrm{\ne }0$, 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 $(P+Q{)}_k=p_k+q_k,(PQ{)}_k={\sum }_{j=0}^k{p}_j{q}_{k-j}\mathrm{.}$ 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 $\mathrm{\Phi }(P,Q):=(Q,2XQ-P)(P,Q\in {\mathcal{P}}_{\mathbb{R}})\mathrm{.}$ 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 $\mathrm{\Phi }$. This gives unique pair sequences $H^T,H^U$. Define $T_n$ and $U_n$ to be the first coordinates of $H_n^T$ and $H_n^U$, respectively. Since the first coordinate of $\mathrm{\Phi }(P,Q)$ is $Q$, the second coordinate of $H_n^T$ is $T_{n+1}$ and the second coordinate of $H_n^U$ is $U_{n+1}$. Consequently $T_0=1,T_1=X,T_{n+2}=2X{T}_{n+1}-T_n,$ $U_0=1,U_1=2X,U_{n+2}=2X{U}_{n+1}-U_n$ for every $n\in \mathbb{N}$. These unique sequences are the Chebyshev polynomials of the first and second kinds.