I've found in a paper the "standard result", without proof, stating that:
$$ \log(1-2yz+z^2)= -2 \sum\limits_{k=1}^\infty \frac{T_k(y)}{k}z^k \quad \text{for all} \quad \vert y\vert\leq 1 \quad \text{and} \quad \vert z\vert<1 $$
Where $T_k$ is the k-th Chebyshev polynomial. My question is can anyone refer to where one can find a proof of this "standard result", or how to go about finding a way to develop an equation of this form for orthogonal polynomials $\{ Q_k \}$.
The generating function of the Chebyshev polynomials of the first kind is $$ \frac{1-yz}{1-2yz+z^2} = \sum_{k=0}^{\infty} z^k T_k(y) $$ (this really is a standard result: it's the first thing mentioned after the recurrence relation on Wikipedia).
The numerator is almost the derivative of the denominator: subtracting $1=T_0(y)$ and dividing by $-z/2$ gives $$ \frac{-2y+2z}{1-2yz+z^2} = -2\sum_{k=1}^{\infty} z^{k-1} T_k(y) , $$ and integrating with respect to $z$ from $0$ gives the result (usual results about absolutely convergent power series, etc.)