Show that first order Chebyshev polynomials are in fact polynomials

Question

Given the 1st order Chebyshev polynomials $$ G(x,t) = \sum_{n=0}^{+\infty} T_n(x) t^n = \frac{1-tx}{1-2xt+t^2} $$

I'm wondering how can I show that $T_n(x)$ are polynomials ?

Answer 1

This just elaborates the hint given by @RandyMarsh. $$1-xt=(1-2xt+t^2)\sum_{n=0}^\infty T_n(x)t^n=\sum_{n=0}^\infty T_n(x)t^n-\sum_{n=0}^\infty 2xT_n(x)t^{n+1}+\sum_{n=0}^\infty T_n(x)t^{n+2}\\=T_0(x)+T_1(x)t+\sum_{n=2}^\infty T_n(x)t^n-2xT_0(x)t-\sum_{n=2}^\infty 2xT_{n-1}(x)t^n+\sum_{n=2}^\infty T_{n-2}(x)t^n\\=T_0(x)+\big(T_1(x)-2xT_0(x)\big)t+\sum_{n=2}^\infty\big(T_n(x)-2xT_{n-1}(x)+T_{n-2}(x)\big)t^n.$$ Now compare the coefficients of $t^0$, $t^1$, and $t^n$ for $n\geqslant 2$. And conclude using induction on $n$.