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## reference request – 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}
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