Difference between revisions of "Jacobi P"

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The Jacobi polynomial $P_n^{(\alpha,\beta)}$ are [[orthogonal polynomials]] defined to be coefficient of $t^n$ in the expansion of
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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 $P_n^{(\alpha,\beta)}(1) = {{n + \alpha} \choose n}$.
$$\dfrac{2^{\alpha+\beta}}{\sqrt{1-2xt+t^2}\left(1-t+ \sqrt{1-2xt+t^2} \right)^{\alpha} \left(1+t+\sqrt{1-2xt+t^2} \right)^{\beta}}$$
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$$P_n^{(\alpha,\beta)}(z)=\dfrac{(\alpha+1)^{\overline{n}}}{n!} {}_2F_1 \left(-n, 1+\alpha+\beta+n;\alpha+1; \dfrac{1}{2}(1-z) \right),$$
in the sense that
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where ${}_2F_1$ is the [[Hypergeometric pFq|generalized hypergeometries series]].
$$\dfrac{2^{\alpha+\beta}}{\sqrt{1-2xt+t^2}\left(1-t+ \sqrt{1-2xt+t^2} \right)^{\alpha} \left(1+t+\sqrt{1-2xt+t^2} \right)^{\beta}} =  \sum_{k=0}^{\infty} P_k^{(\alpha,\beta)}(x)t^k$$
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holds.
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=Properties=
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[[Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials]]<br />
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[[Differential equation for Jacobi P]]<br />
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=References=
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* {{BookReference|Orthogonal Polynomials|1975|Gabor Szegő|edpage = Fourth Edition|prev=findme|next=findme}}: page 58
  
 
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{{:Orthogonal polynomials footer}}
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[[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

Orthogonal polynomials