Difference between revisions of "Jacobi P"
From specialfunctionswiki
(\lef) |
|||
| (7 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | The Jacobi polynomial $P_n^{(\alpha,\beta)}$ are | + | 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 $P_n^{(\alpha,\beta)}(1) = {{n + \alpha} \choose n}$. |
$$P_n^{(\alpha,\beta)}(z)=\dfrac{(\alpha+1)^{\overline{n}}}{n!} {}_2F_1 \left(-n, 1+\alpha+\beta+n;\alpha+1; \dfrac{1}{2}(1-z) \right),$$ | $$P_n^{(\alpha,\beta)}(z)=\dfrac{(\alpha+1)^{\overline{n}}}{n!} {}_2F_1 \left(-n, 1+\alpha+\beta+n;\alpha+1; \dfrac{1}{2}(1-z) \right),$$ | ||
where ${}_2F_1$ is the [[Hypergeometric pFq|generalized hypergeometries series]]. | where ${}_2F_1$ is the [[Hypergeometric pFq|generalized hypergeometries series]]. | ||
=Properties= | =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}} | {{: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
