Difference between revisions of "Jacobi P"
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| − | The Jacobi polynomial $P_n^{(\alpha,\beta)}$ are | + | The Jacobi polynomial $P_n^{(\alpha,\beta)}$ are defined by |
| − | $$\dfrac{ | + | $$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),$$ |
| − | + | where ${}_2F_1$ is the [[Hypergeometric pFq|generalized hypergeometries series]]. | |
| − | $$ | + | |
| − | + | =Properties= | |
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> ([[Rodrigues' formula]]) The following formula holds: | ||
| + | $$P_n^{(\alpha,\beta)}(z)=\dfrac{(-1)^n}{2^nn!} (1-z)^{-\alpha}(1+z)^{-\beta} \dfrac{d^n}{dz^n} \left[(1-z)^{\alpha}(1+z)^{\beta}(1-z^2)^n \right].$$ | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
{{:Orthogonal polynomials footer}} | {{:Orthogonal polynomials footer}} | ||
Revision as of 20:17, 23 March 2015
The Jacobi polynomial $P_n^{(\alpha,\beta)}$ are defined by $$P_n^{(\alpha,\beta)}(z)=\dfrac{(\alpha+1)^{\overline{n}}}{n!} {}_2F_1 \left(-n, 1+\alpha+\beta+n;\alpha+1; \dfrac{1}{2}(1-z) \right),$$ where ${}_2F_1$ is the generalized hypergeometries series.
Properties
Theorem: (Rodrigues' formula) The following formula holds: $$P_n^{(\alpha,\beta)}(z)=\dfrac{(-1)^n}{2^nn!} (1-z)^{-\alpha}(1+z)^{-\beta} \dfrac{d^n}{dz^n} \left[(1-z)^{\alpha}(1+z)^{\beta}(1-z^2)^n \right].$$
Proof: █
