Difference between revisions of "Gegenbauer C"
| Line 1: | Line 1: | ||
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},$$ |
where $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the [[floor]] function, $\Gamma$ denotes [[gamma]], and $k!$ denotes the [[factorial]]. | where $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the [[floor]] function, $\Gamma$ denotes [[gamma]], and $k!$ denotes the [[factorial]]. | ||
| Line 8: | Line 8: | ||
[[nC_n^(lambda)(x)=2lambda(xC_(n-1)^(lambda+1)(x)-C_(n-2)^(lambda+1)(x))]]<br /> | [[nC_n^(lambda)(x)=2lambda(xC_(n-1)^(lambda+1)(x)-C_(n-2)^(lambda+1)(x))]]<br /> | ||
[[(n+2lambda)C_n^(lambda)(x)=2lambda(C_n^(lambda+1)(x)-xC_(n-1)^(lambda+1)(x))]]<br /> | [[(n+2lambda)C_n^(lambda)(x)=2lambda(C_n^(lambda+1)(x)-xC_(n-1)^(lambda+1)(x))]]<br /> | ||
| + | [[nC_n^(lambda)(x)=(n-1+2lambda)xC_(n-1)^(lambda)(x)-2lambda(1-x^2)C_(n-2)^(lambda-1)(x)]]<br /> | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | <div class="toccolours mw-collapsible mw-collapsed" style="width:800px"> | ||
Revision as of 01:27, 20 December 2017
The Gegenbauer polynomial $C_n^{\lambda}$ of degree $n \in \{0,1,2,\ldots\}$ and order $\lambda$ defined by $$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},$$ where $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the floor function, $\Gamma$ denotes gamma, and $k!$ denotes the factorial.
Properties
Orthogonality of Gegenbauer C on (-1,1)
(n+2)C_(n+2)^(lambda)(x)=2(lambda+n+1)xC_(n+1)^(lambda)(x)-(2lambda+n)C_n^(lambda)(x)
nC_n^(lambda)(x)=2lambda(xC_(n-1)^(lambda+1)(x)-C_(n-2)^(lambda+1)(x))
(n+2lambda)C_n^(lambda)(x)=2lambda(C_n^(lambda+1)(x)-xC_(n-1)^(lambda+1)(x))
nC_n^(lambda)(x)=(n-1+2lambda)xC_(n-1)^(lambda)(x)-2lambda(1-x^2)C_(n-2)^(lambda-1)(x)
Theorem: The following formula holds: $$C_n^{\lambda '}(x) = 2\lambda C_{n+1}^{\lambda+1}(x).$$
Proof: █
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
