Difference between revisions of "Gegenbauer C"

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The Gegenbauer polynomial of degree $n$ and order $\lambda$ are [[orthogonal polynomials]] defined to be the coefficient of $t^n$ in the expansion of
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The Gegenbauer polynomial $C_n^{\lambda}$ of degree $n \in \{0,1,2,\ldots\}$ and order $\lambda$ defined by
$\dfrac{1}{(1-2xt+t^2)^{\lambda}}$ in the sense that
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$$C_n^{\lambda}(x)=\displaystyle\sum_{k=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \dfrac{(-1)^k\Gamma(n-k+\lambda)}{\Gamma(\lambda)k!(n-2k)!} (2x)^{n-2k},$$
$$\dfrac{1}{(1-2xt+t^2)^{\lambda}} = \sum_{k=0}^{\infty} C_k^{\lambda}(x)t^k.$$
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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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[[Orthogonality of Gegenbauer C on (-1,1)]]<br />
<strong>Theorem:</strong> The Gegenbauer polynomials satisfy the differential equation
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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 />
$$(1-x^2)\dfrac{d^2y}{dx^2} -(2\lambda+1) x \dfrac{dy}{dx} + n(n+2\lambda)y=0.$$
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[[nC_n^(lambda)(x)=2lambda(xC_(n-1)^(lambda+1)(x)-C_(n-2)^(lambda+1)(x))]]<br />
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[[(n+2lambda)C_n^(lambda)(x)=2lambda(C_n^(lambda+1)(x)-xC_(n-1)^(lambda+1)(x))]]<br />
<strong>Proof:</strong>
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[[nC_n^(lambda)(x)=(n-1+2lambda)xC_(n-1)^(lambda)(x)-2lambda(1-x^2)C_(n-2)^(lambda-1)(x)]]<br />
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[[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 />
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[[Relationship between Chebyshev T and Gegenbauer C]]<br />
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[[Relationship between Chebyshev U and Gegenbauer C]]<br />
  
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{{:Orthogonal polynomials footer}}
<strong>Theorem:</strong> The following formula holds:
 
$$C_n^{\lambda}(x) = \displaystyle\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} (-1)^k \dfrac{\Gamma(n-k+\lambda)}{\Gamma(\lambda)k!(n-2k)!} (2x)^{n-2k}.$$
 
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<strong>Proof:</strong> █
 
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</div>
 
  
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[[Category:SpecialFunction]]
<strong>Theorem (Orthogonality):</strong> The following formula holds:
 
$$\displaystyle\int_{-1}^1 (1-x^2)^{\lambda-\frac{1}{2}} C_n^{\lambda}(x)C_m^{\lambda}(x) dx = 2^{1-2\lambda} \pi \dfrac{\Gamma(n+2\lambda)}{(n+\lambda)(\Gamma(\lambda))^2\Gamma(n+1)}\delta_{mn},$$
 
where $\delta_{mn}=0$ if $m\neq 0$ and $\delta_{mn}=1$ when $m=n$.
 
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<strong>Proof:</strong> █
 
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</div>
 
 
 
 
 
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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> █
 
</div>
 
</div>
 
 
 
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<strong>Theorem:</strong> The following formula holds:
 
$$nC_n^{\lambda}(x) = 2\lambda(xC_{n-1}^{\lambda+1}(x) - C_{n-2}^{\lambda+1}(x)).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
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<strong>Theorem:</strong> The following formula holds:
 
$$(n+2\lambda)C_n^{\lambda}(x) = 2\lambda(C_n^{\lambda+1}(x)-xC_{n-1}^{\lambda+1}(x))$$.
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
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<strong>Theorem:</strong> The following formula holds:
 
$$nC_n^{\lambda}(x) = (n-1+2\lambda)xC_{n-1}^{\lambda}(x) - 2\lambda(1-x^2)C_{n-2}^{\lambda-1}(x).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 
 
 
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<strong>Theorem:</strong> The following formula holds:
 
$$C_n^{\lambda '}(x) = 2\lambda C_{n+1}^{\lambda+1}(x).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong> █
 
</div>
 
</div>
 

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

Orthogonal polynomials