Difference between revisions of "Jacobi P"
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(Created page with "The Jacobi polynomial $P_n^{(\alpha,\beta)}$ is the coefficient of $t^n$ in the expansion of $$\dfrac{2^{\alpha+\beta}}{\sqrt{1-2xt+t^2}\left(1-t+ \sqrt{1-2xt+t^2} \right)^{\...") |
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| − | The Jacobi polynomial $P_n^{(\alpha,\beta)}$ | + | Let $\alpha > -1$ and $\beta > -1$. The Jacobi polynomial $P_n^{(\alpha,\beta)}$ are [[orthogonal polynomials]] with [[weight function]] $w(x)=(1-x)^{\alpha}(1-x)^{\beta}$ on the interval $[-1,1]$ that obey $P_n^{(\alpha,\beta)}(1) = {{n + \alpha} \choose n}$. |
| − | + | $$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),$$ | |
| − | + | where ${}_2F_1$ is the [[Hypergeometric pFq|generalized hypergeometries series]]. | |
| − | + | ||
| − | + | =Properties= | |
| + | [[Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials]]<br /> | ||
| + | [[Differential equation for Jacobi P]]<br /> | ||
| + | |||
| + | =References= | ||
| + | * {{BookReference|Orthogonal Polynomials|1975|Gabor Szegő|edpage = Fourth Edition|prev=findme|next=findme}}: page 58 | ||
| + | |||
| + | {{:Orthogonal polynomials footer}} | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 03:30, 11 June 2016
Let $\alpha > -1$ and $\beta > -1$. The Jacobi polynomial $P_n^{(\alpha,\beta)}$ are orthogonal polynomials with weight function $w(x)=(1-x)^{\alpha}(1-x)^{\beta}$ on the interval $[-1,1]$ that obey $P_n^{(\alpha,\beta)}(1) = {{n + \alpha} \choose n}$. $$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
- 1975: Gabor Szegő: Orthogonal Polynomials ... (previous) ... (next): page 58
