The generating function for Chebyshev polynomials of the first kind is given as:
$$\sum_{n=0}^{\infty}T_n(x)t^n=\frac{1-xt}{1-2xt+t^2}$$
As I understand it, the sum of the coefficients of the generating function for Chebyshev polynomials is given by letting $t=1$. This seems to be clear from inspecting the LHS of the equation. However, the RHS of the equation simply reduces to $\frac{1}{2}$ and is no longer a function of $x$. Can anybody resolve this apparent anomaly? Thank you in advance!
There is this $|t|<1$ restriction for this formula to be valid (the lhs series must be convergent)
The proof can be found page 116 of https://eprints.ucm.es/32785/1/T36277.pdf
The main steps are: $$ T_n(x) = \cos(n \arccos(x)) = \Re[\exp(i\,n\arccos(x))] $$ which is valid for any $x\in\mathbb{R}$
However, if $|t|<1$ $$ \sum_{n=0}^\infty t^n \exp(i\,n\arccos(x))=\sum_{n=0}^\infty (t \exp(i \arccos(x)))^n=\frac{1}{1-t\exp(i \arccos(x))} $$
Taking the real part of this gives you your generating function