Difference between revisions of "T n(x)=Sum (-1)^k n!/((2k)! (n-2k)!) (1-x^2)^k x^(n-2k)"
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(Created page with "==Theorem== The following formula holds: $$T_n(x) = \displaystyle\sum_{k=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \dfrac{(-1)^k n!}{(2k)!(n-2k)!} (1-x^2)^k x^{n-2k},$$ wher...") |
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==References== | ==References== | ||
| − | * {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=U n(x)=(-i/2)(x+i sqrt(1-x^2))^n+(-i/2)(x-i sqrt(1-x^2))^n|next= | + | * {{BookReference|Special Functions for Scientists and Engineers|1968|W.W. Bell|prev=U n(x)=(-i/2)(x+i sqrt(1-x^2))^n+(-i/2)(x-i sqrt(1-x^2))^n|next=U n(x)=Sum (-1)^k n!/((2k+1)!(n-2k-1)!)(1-x^2)^(k+1/2)x^(n-2k-1)}}: Theorem 7.2 (i) |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 19:28, 15 March 2018
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.
Proof
References
- 1968: W.W. Bell: Special Functions for Scientists and Engineers ... (previous) ... (next): Theorem 7.2 (i)
