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| | =Properties= | | =Properties= |
| − | <div class="toccolours mw-collapsible mw-collapsed">
| + | [[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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| − | $$P_n^{(\alpha,\beta)}(z)=\dfrac{(-1)^n}{2^nn!} (1-z)^{-\alpha}(1+z)^{-\beta} \dfrac{d^n}{dz^n} \left[(1-z)^{\alpha}(1+z)^{\beta}(1-z^2)^n \right].$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | <div class="toccolours mw-collapsible mw-collapsed">
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| − | <strong>Theorem:</strong> (Orthogonality) The following formula holds:
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| − | $$\displaystyle\int_{-1}^1 (-1-x)^{\alpha}(1+x)^{\beta}P_n^{(\alpha,\beta)}(x)P_m^{(\alpha,\beta)}(x)dx=\dfrac{2^{\alpha+\beta+1}\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{(2n+\alpha+\beta+1)n! \Gamma(n+\alpha+\beta+1)}\delta_{mn},$$
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| − | where $\delta_{mn}$ denotes the [[Dirac delta]].
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | <div class="toccolours mw-collapsible mw-collapsed">
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| − | <strong>Theorem:</strong> The $P_n^{(\alpha,\beta)}$ functions satisfy the differential equation
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| − | $$(1-x^2)\dfrac{d^2y}{dx^2}+(\beta-\alpha-(\alpha+\beta+2)x)\dfrac{dy}{dx}+n(n+\alpha+\beta+1)y=0.$$
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| − | <div class="mw-collapsible-content">
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| − | <strong>Proof:</strong> █
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| − | </div>
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| − | </div>
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| − | {{:Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials}}
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| | =References= | | =References= |
Revision as of 03:28, 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
References
Orthogonal polynomials
Jacobi $P^{(\alpha,\beta)}$
