Difference between revisions of "Relationship between Chebyshev T and hypergeometric 2F1"
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| − | + | ==Theorem== | |
| − | + | The following formula holds for $n \in \{0,1,2,\ldots\}$: | |
$$T_n(x) = {}_2F_1 \left( -n,n ; \dfrac{1}{2}; \dfrac{1-x}{2} \right),$$ | $$T_n(x) = {}_2F_1 \left( -n,n ; \dfrac{1}{2}; \dfrac{1-x}{2} \right),$$ | ||
where $T_n$ denotes a [[Chebyshev T]] polynomial and ${}_2F_1$ denotes the [[hypergeometric pFq]]. | where $T_n$ denotes a [[Chebyshev T]] polynomial and ${}_2F_1$ denotes the [[hypergeometric pFq]]. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
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 T polynomial and ${}_2F_1$ denotes the hypergeometric pFq.
