Difference between revisions of "Gegenbauer C"
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| − | The Gegenbauer polynomial | + | The Gegenbauer polynomial $C_n^{\lambda}$ of degree $n \in \{0,1,2,\ldots\}$ and order $\lambda$ defined by |
| − | + | $$C_n^{\lambda}(z)=\displaystyle\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \dfrac{(-1)^k\Gamma(n-k+\lambda)}{\Gamma(\lambda)k!(n-2k)!} (2z)^{n-2k},$$ | |
| − | $$ | + | where $\lfloor \frac{n}{2} \rfloor$ denotes the [[floor]] function, $\Gamma$ denotes [[gamma]], and $k!$ denotes the [[factorial]]. |
=Properties= | =Properties= | ||
| − | + | [[Orthogonality of Gegenbauer C on (-1,1)]]<br /> | |
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Revision as of 23:42, 19 December 2017
The Gegenbauer polynomial $C_n^{\lambda}$ of degree $n \in \{0,1,2,\ldots\}$ and order $\lambda$ defined by $$C_n^{\lambda}(z)=\displaystyle\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \dfrac{(-1)^k\Gamma(n-k+\lambda)}{\Gamma(\lambda)k!(n-2k)!} (2z)^{n-2k},$$ where $\lfloor \frac{n}{2} \rfloor$ denotes the floor function, $\Gamma$ denotes gamma, and $k!$ denotes the factorial.
Properties
Orthogonality of Gegenbauer C on (-1,1)
Theorem: The following formula holds: $$(n+2)C_{n+2}^{\lambda}(x)=2(\lambda+n+1)xC_{n+1}^{\lambda}(x)-(2\lambda+n)C_n^{\lambda}(x).$$
Proof: █
Theorem: The following formula holds: $$nC_n^{\lambda}(x) = 2\lambda(xC_{n-1}^{\lambda+1}(x) - C_{n-2}^{\lambda+1}(x)).$$
Proof: █
Theorem: The following formula holds: $$(n+2\lambda)C_n^{\lambda}(x) = 2\lambda(C_n^{\lambda+1}(x)-xC_{n-1}^{\lambda+1}(x))$$.
Proof: █
Theorem: The following formula holds: $$nC_n^{\lambda}(x) = (n-1+2\lambda)xC_{n-1}^{\lambda}(x) - 2\lambda(1-x^2)C_{n-2}^{\lambda-1}(x).$$
Proof: █
Theorem: The following formula holds: $$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
