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

Theorem

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

References

• 1968: W.W. Bell: Special Functions for Scientists and Engineers ... (previous) ... (next): Theorem 7.2 (i)