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W. Wang
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Resultant of linear combinations of Chebyshev polynomials of the second kind

The Chebyshev polynomial $U_n(x)$ of the second kind is characterized by $U_n(cos\theta)=\frac{\sin(n+1)\theta}{\sin(\theta)}$. It seems that $$Res_x(U_n(x)+tU_{n-1}(x),\sum_{k=0}^{n-1}U_k(x))=(-1)^{\frac{n(n-1)}{2}}t^{\lfloor\frac{k}{2}\rfloor}2^{n(n-1)},$$ where Res denotes the resultant of two polynomials.

I do not know how to prove this "equality". Is it a known result? A similar result appeared in https://www.researchgate.net/publication/228924014_Resultants_and_discriminants_of_Chebyshev_and_related_polynomials, Theorem 2.

W. Wang
  • 437
  • 2
  • 6