# Index-decomposition identity for Chebyshev polynomials

Let $$n$$ be a positive integer. Denote $$\left[ n \right] \equiv \{1, \ldots , n \}$$. Denote $$T_{n} \left( x \right)$$ as the $$n$$-th Chebyshev polynomials of the first kind. Let $$m_{1}, \ldots , m_{n}$$ be positive integers. Then an "index-decomposition" identity for the Chebyshev polynomial of the first kind is given by \begin{align} T_{m_{1} + \dots + m_{n}} \left( x \right) & = \sum_{\begin{aligned} I \subseteq \left[ n \right] \, \\ \lvert I \rvert \, \text{even} \end{aligned}}{ \left( -1 \right)^{\frac{\lvert I \rvert}{2}} \prod_{i \in I}{\sqrt{1-T_{m_{i}} \left( x \right)^{2}}} \prod_{j \in \left[ n\right] \setminus I}{T_{m_{j}} \left( x \right)}} \\ & = \sum_{\begin{aligned} I \subseteq \left[ n \right] \, \\ \lvert I \rvert \, \text{even} \end{aligned}}{ \left( \frac{T_{2} \left( x \right) -1}{2} \right)^{\frac{\lvert I \rvert}{2}} \prod_{i \in I}{ \frac{T_{m_{i}} \left( x \right)'}{m_{i}}} \prod_{j \in \left[ n\right] \setminus I}{T_{m_{j}} \left( x \right)}} \end{align} This identity came up in the context of a previous question.

I am asking for references to the identity; preferably those which also consist of applications and/or generalizations.

• If you set $x=\cos\theta$, this identity is a generalization of the formula $\cos(m\theta+n\theta) = \cos(m\theta)\cos(n\theta) - \sin(m\theta)\sin(n\theta)$. Commented Jun 23, 2022 at 19:54