U n(x)=Sum (-1)^k n!/((2k+1)!(n-2k-1)!)(1-x^2)^(k+1/2)x^(n-2k-1)

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Theorem

The following formula holds: $$U_n(x) = \displaystyle\sum_{k=0}^{\left\lfloor \frac{n-1}{2} \right\rfloor} \dfrac{(-1)^k n!}{(2k+1)!(n-2k-1)!} (1-x^2)^{k+\frac{1}{2}}x^{n-2k-1},$$ where $U_n$ denotes Chebyshev U and $\lfloor \frac{n-1}{2} \rfloor$ denotes the floor.

Proof

References