U n(x)=Sum (-1)^k n!/((2k+1)!(n-2k-1)!)(1-x^2)^(k+1/2)x^(n-2k-1)
From specialfunctionswiki
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
- 1968: W.W. Bell: Special Functions for Scientists and Engineers ... (previous) ... (next): Theorem 7.2 (ii)
