Difference between revisions of "Relationship between Chebyshev T and Gegenbauer C"
From specialfunctionswiki
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==Theorem== | ==Theorem== | ||
| − | The following formula holds for $ | + | The following formula holds for $n \in \{1,2,3,\ldots\}$: |
| − | $$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda} | + | $$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda},$$ |
where $T_n$ denotes a [[Chebyshev T|Chebyshev polynomial of the first kind]] and $C_n^{\lambda}$ denotes a [[Gegenbauer C]] polynomial. | where $T_n$ denotes a [[Chebyshev T|Chebyshev polynomial of the first kind]] and $C_n^{\lambda}$ denotes a [[Gegenbauer C]] polynomial. | ||
Latest revision as of 22:33, 19 December 2017
Theorem
The following formula holds for $n \in \{1,2,3,\ldots\}$: $$T_n(x)=\dfrac{n}{2} \displaystyle\lim_{\lambda \rightarrow 0} \dfrac{C_n^{\lambda}(x)}{\lambda},$$ where $T_n$ denotes a Chebyshev polynomial of the first kind and $C_n^{\lambda}$ denotes a Gegenbauer C polynomial.
