Anyway, I think Chebyshev Polynomials of the First Kind are done wrong. I want to introduce the polynomial that generates these polynomials.
T_n shares all the properties that Lucas Numbers and Polynomials have. U_n shares all the properties that Fibonacci Numbers and Polynomials have. Well, not exactly. Since it's algebra, the impact is often algebraic. Calculating the sums, for example, is different. Also, I would like to point up that as you go higher in degrees, you gain more Fibonacci/Lucas variants. There are infinite variants in degree two, but they seem to lack meaning. These extra variants that you gain, as you increase the degree, have meaning.