Difference between revisions of "Gegenbauer C"

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(Properties)
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=Properties=
 
=Properties=
 
[[Orthogonality of Gegenbauer C on (-1,1)]]<br />
 
[[Orthogonality of Gegenbauer C on (-1,1)]]<br />
 
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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 />
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<strong>Theorem:</strong> The following formula holds:
 
$$(n+2)C_{n+2}^{\lambda}(x)=2(\lambda+n+1)xC_{n+1}^{\lambda}(x)-(2\lambda+n)C_n^{\lambda}(x).$$
 
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<strong>Proof:</strong> █
 
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Revision as of 23:46, 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)
(n+2)C_(n+2)^(lambda)(x)=2(lambda+n+1)xC_(n+1)^(lambda)(x)-(2lambda+n)C_n^(lambda)(x)

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

Orthogonal polynomials