Difference between revisions of "Relationship between Chebyshev U and hypergeometric 2F1"
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| − | + | ==Theorem== | |
| − | + | The following formula holds for $n \in \{0,1,2,\ldots\}$: | |
$$U_n(x) = (n+1){}_2F_1 \left( -n,n+2 ; \dfrac{3}{2}; \dfrac{1-x}{2} \right),$$ | $$U_n(x) = (n+1){}_2F_1 \left( -n,n+2 ; \dfrac{3}{2}; \dfrac{1-x}{2} \right),$$ | ||
| − | where $U_n$ denotes a [[Chebyshev U]] | + | where $U_n$ denotes a [[Chebyshev U|Chebyshev polynomial of the second kind]] and ${}_2F_1$ denotes [[hypergeometric 2F1]]. |
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 22:43, 19 December 2017
Theorem
The following formula holds for $n \in \{0,1,2,\ldots\}$: $$U_n(x) = (n+1){}_2F_1 \left( -n,n+2 ; \dfrac{3}{2}; \dfrac{1-x}{2} \right),$$ where $U_n$ denotes a Chebyshev polynomial of the second kind and ${}_2F_1$ denotes hypergeometric 2F1.
