Difference between revisions of "Relationship between Chebyshev T and hypergeometric 2F1"
From specialfunctionswiki
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The following formula holds for $n \in \{0,1,2,\ldots\}$: | The following formula holds for $n \in \{0,1,2,\ldots\}$: | ||
$$T_n(x) = {}_2F_1 \left( -n,n ; \dfrac{1}{2}; \dfrac{1-x}{2} \right),$$ | $$T_n(x) = {}_2F_1 \left( -n,n ; \dfrac{1}{2}; \dfrac{1-x}{2} \right),$$ | ||
| − | where $T_n$ denotes a [[Chebyshev T]] | + | where $T_n$ denotes a [[Chebyshev T|Chebyshev polynomial of the first kind]] and ${}_2F_1$ denotes the [[hypergeometric pFq]]. |
==Proof== | ==Proof== | ||
Latest revision as of 22:32, 19 December 2017
Theorem
The following formula holds for $n \in \{0,1,2,\ldots\}$: $$T_n(x) = {}_2F_1 \left( -n,n ; \dfrac{1}{2}; \dfrac{1-x}{2} \right),$$ where $T_n$ denotes a Chebyshev polynomial of the first kind and ${}_2F_1$ denotes the hypergeometric pFq.
