The following is similar to one of the generating functions for Chebyshev polynomials but differs in the summation limits ranging from $-\infty$ to $+\infty$ rather than $0$ to $+\infty$:
$\sum\limits_{n=-\infty}^{+\infty} T_n(\tau)\frac{e^{jn\alpha}}{n}$.
The wikipdeia page for Chebyshev polynomials has the following identity for the one-sided summation:
$\sum\limits_{n=0}^{+\infty} T_n(x)\frac{t^n}{n} = \ln{ \frac{e}{\sqrt{ 1 - 2tx + t^2 } } } $.
Is the above valid for $t$ a complex exponential? And how would I go about deriving a similar identity for the case of $n=(-\infty,+\infty)$?
Edit: Is the above formula even correct? It looks to be undefined for $n=0$.
