Difference between revisions of "Jacobi P"

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=Properties=
 
=Properties=
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[[Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials]]<br />
<strong>Theorem:</strong> ([[Rodrigues' formula]]) The following formula holds:
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[[Differential equation for Jacobi P]]<br />
$$P_n^{(\alpha,\beta)}(z)=\dfrac{(-1)^n}{2^nn!} (1-z)^{-\alpha}(1+z)^{-\beta} \dfrac{d^n}{dz^n} \left[(1-z)^{\alpha}(1+z)^{\beta}(1-z^2)^n \right].$$
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> (Orthogonality) The following formula holds:
 
$$\displaystyle\int_{-1}^1 (-1-x)^{\alpha}(1+x)^{\beta}P_n^{(\alpha,\beta)}(x)P_m^{(\alpha,\beta)}(x)dx=\dfrac{2^{\alpha+\beta+1}\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{(2n+\alpha+\beta+1)n! \Gamma(n+\alpha+\beta+1)}\delta_{mn},$$
 
where $\delta_{mn}$ denotes the [[Dirac delta]].
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The $P_n^{(\alpha,\beta)}$ functions satisfy the differential equation
 
$$(1-x^2)\dfrac{d^2y}{dx^2}+(\beta-\alpha-(\alpha+\beta+2)x)\dfrac{dy}{dx}+n(n+\alpha+\beta+1)y=0.$$
 
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<strong>Proof:</strong> █
 
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{{:Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials}}
 
  
 
=References=
 
=References=

Latest revision as of 03:30, 11 June 2016

Let $\alpha > -1$ and $\beta > -1$. The Jacobi polynomial $P_n^{(\alpha,\beta)}$ are orthogonal polynomials with weight function $w(x)=(1-x)^{\alpha}(1-x)^{\beta}$ on the interval $[-1,1]$ that obey $P_n^{(\alpha,\beta)}(1) = {{n + \alpha} \choose n}$. $$P_n^{(\alpha,\beta)}(z)=\dfrac{(\alpha+1)^{\overline{n}}}{n!} {}_2F_1 \left(-n, 1+\alpha+\beta+n;\alpha+1; \dfrac{1}{2}(1-z) \right),$$ where ${}_2F_1$ is the generalized hypergeometries series.

Properties

Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials
Differential equation for Jacobi P

References

Orthogonal polynomials