Difference between revisions of "Relationship between Chebyshev U and Gegenbauer C"
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| − | + | ==Theorem== | |
| − | + | The following formula holds for $n \in \{1,2,3,\ldots\}$: | |
$$U_n(x)=\sqrt{1-x^2}C_{n-1}^1(x),$$ | $$U_n(x)=\sqrt{1-x^2}C_{n-1}^1(x),$$ | ||
| − | where $U_n$ denotes a [[Chebyshev U]] | + | where $U_n$ denotes a [[Chebyshev U|Chebyshev polynomial of the second kind]] and $C_{n-1}^1$ denotes a [[Gegenbauer C]] polynomial. |
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 22:49, 19 December 2017
Theorem
The following formula holds for $n \in \{1,2,3,\ldots\}$: $$U_n(x)=\sqrt{1-x^2}C_{n-1}^1(x),$$ where $U_n$ denotes a Chebyshev polynomial of the second kind and $C_{n-1}^1$ denotes a Gegenbauer C polynomial.
