Difference between revisions of "Relationship between Chebyshev T and Gegenbauer C"
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| − | + | ==Theorem== | |
| − | + | The following formula holds for $m,n \in \{0,1,2,\ldots\}$: | |
$$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda}; n\geq 1,$$ | $$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda}; n\geq 1,$$ | ||
| − | where $T_n$ denotes a [[Chebyshev T]] | + | where $T_n$ denotes a [[Chebyshev T|Chebyshev polynomial of the first kind]] and $C_n^{\lambda}$ denotes a [[Gegenbauer C]] polynomial. |
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Revision as of 22:33, 19 December 2017
Theorem
The following formula holds for $m,n \in \{0,1,2,\ldots\}$: $$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda}; n\geq 1,$$ where $T_n$ denotes a Chebyshev polynomial of the first kind and $C_n^{\lambda}$ denotes a Gegenbauer C polynomial.
