Difference between revisions of "Gegenbauer C"
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The Gegenbauer polynomial $C_n^{\lambda}$ of degree $n \in \{0,1,2,\ldots\}$ and order $\lambda$ defined by | The Gegenbauer polynomial $C_n^{\lambda}$ of degree $n \in \{0,1,2,\ldots\}$ and order $\lambda$ defined by | ||
| − | $$C_n^{\lambda}( | + | $$C_n^{\lambda}(x)=\displaystyle\sum_{k=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \dfrac{(-1)^k\Gamma(n-k+\lambda)}{\Gamma(\lambda)k!(n-2k)!} (2x)^{n-2k},$$ |
| − | where $\lfloor \frac{n}{2} \rfloor$ denotes the [[floor]] function, $\Gamma$ denotes [[gamma]], and $k!$ denotes the [[factorial]]. | + | where $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the [[floor]] function, $\Gamma$ denotes [[gamma]], and $k!$ denotes the [[factorial]]. |
=Properties= | =Properties= | ||
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[[(n+2)C_(n+2)^(lambda)(x)=2(lambda+n+1)xC_(n+1)^(lambda)(x)-(2lambda+n)C_n^(lambda)(x)]]<br /> | [[(n+2)C_(n+2)^(lambda)(x)=2(lambda+n+1)xC_(n+1)^(lambda)(x)-(2lambda+n)C_n^(lambda)(x)]]<br /> | ||
[[nC_n^(lambda)(x)=2lambda(xC_(n-1)^(lambda+1)(x)-C_(n-2)^(lambda+1)(x))]]<br /> | [[nC_n^(lambda)(x)=2lambda(xC_(n-1)^(lambda+1)(x)-C_(n-2)^(lambda+1)(x))]]<br /> | ||
| − | + | [[(n+2lambda)C_n^(lambda)(x)=2lambda(C_n^(lambda+1)(x)-xC_(n-1)^(lambda+1)(x))]]<br /> | |
| − | + | [[nC_n^(lambda)(x)=(n-1+2lambda)xC_(n-1)^(lambda)(x)-2lambda(1-x^2)C_(n-2)^(lambda-1)(x)]]<br /> | |
| − | + | [[C_n^(lambda)'(x)=2lambda C_(n+1)^(lambda+1)(x)]]<br /> | |
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[[Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials]]<br /> | [[Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials]]<br /> | ||
[[Relationship between Chebyshev T and Gegenbauer C]]<br /> | [[Relationship between Chebyshev T and Gegenbauer C]]<br /> | ||
Latest revision as of 01:29, 20 December 2017
The Gegenbauer polynomial $C_n^{\lambda}$ of degree $n \in \{0,1,2,\ldots\}$ and order $\lambda$ defined by $$C_n^{\lambda}(x)=\displaystyle\sum_{k=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \dfrac{(-1)^k\Gamma(n-k+\lambda)}{\Gamma(\lambda)k!(n-2k)!} (2x)^{n-2k},$$ where $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the floor function, $\Gamma$ denotes gamma, and $k!$ denotes the factorial.
Properties
Orthogonality of Gegenbauer C on (-1,1)
(n+2)C_(n+2)^(lambda)(x)=2(lambda+n+1)xC_(n+1)^(lambda)(x)-(2lambda+n)C_n^(lambda)(x)
nC_n^(lambda)(x)=2lambda(xC_(n-1)^(lambda+1)(x)-C_(n-2)^(lambda+1)(x))
(n+2lambda)C_n^(lambda)(x)=2lambda(C_n^(lambda+1)(x)-xC_(n-1)^(lambda+1)(x))
nC_n^(lambda)(x)=(n-1+2lambda)xC_(n-1)^(lambda)(x)-2lambda(1-x^2)C_(n-2)^(lambda-1)(x)
C_n^(lambda)'(x)=2lambda C_(n+1)^(lambda+1)(x)
Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials
Relationship between Chebyshev T and Gegenbauer C
Relationship between Chebyshev U and Gegenbauer C
