Jacobi polynomials

The Jacobi polynomials are defined via the hypergeometric function as follows:{{Abramowitz_Stegun_ref|22|561}}{{Pg|location=IV.1}}

:$P\_n^\{(\backslash alpha,\backslash beta)\}(z)=\backslash frac\{(\backslash alpha+1)\_n\}\{n!\}\backslash ,\{\}\_2F\_1\backslash left(-n,1+\backslash alpha+\backslash beta+n;\backslash alpha+1;\backslash tfrac\{1\}\{2\}(1-z)\backslash right),$

where $(\backslash alpha+1)\_n$ is Pochhammer's symbol (for the rising factorial). In this case, the series for the hypergeometric function is finite, therefore one obtains the following equivalent expression:

:$P\_n^\{(\backslash alpha,\backslash beta)\}\; (z)\; =\; \backslash frac\{\backslash Gamma\; (\backslash alpha+n+1)\}\{n!\backslash ,\backslash Gamma\; (\backslash alpha+\backslash beta+n+1)\}\; \backslash sum\_\{m=0\}^n\; \{n\backslash choose\; m\}\; \backslash frac\{\backslash Gamma\; (\backslash alpha\; +\; \backslash beta\; +\; n\; +\; m\; +\; 1)\}\{\backslash Gamma\; (\backslash alpha\; +\; m\; +\; 1)\}\; \backslash left(\backslash frac\{z-1\}\{2\}\backslash right)^m.$

An equivalent definition is given by Rodrigues' formula:{{Pg|location=IV.3}}{{SpringerEOM|title=Jacobi polynomials|author=P.K. Suetin}}

:$P\_n^\{(\backslash alpha,\backslash beta)\}(z)\; =\; \backslash frac\{(-1)^n\}\{2^n\; n!\}\; (1-z)^\{-\backslash alpha\}\; (1+z)^\{-\backslash beta\}\; \backslash frac\{d^n\}\{dz^n\}\; \backslash left\backslash \{\; (1-z)^\backslash alpha\; (1+z)^\backslash beta\; \backslash left\; (1\; -\; z^2\; \backslash right\; )^n\; \backslash right\backslash \}.$

If $\backslash alpha\; =\; \backslash beta\; =\; 0$, then it reduces to the Legendre polynomials:

:$P\_\{n\}(z)\; =\; \backslash frac\{1\; \}\{2^n\; n!\; \}\; \backslash frac\{d^n\; \}\{\; d\; z^n\; \}\; (\; z^2\; -\; 1\; )^n\; \backslash ;\; .$

For real $x$ the Jacobi polynomial can alternatively be written as

:$P\_n^\{(\backslash alpha,\backslash beta)\}(x)=\; \backslash sum\_\{s=0\}^n\; \{n+\backslash alpha\backslash choose\; n-s\}\{n+\backslash beta\; \backslash choose\; s\}\; \backslash left(\backslash frac\{x-1\}\{2\}\backslash right)^\{s\}\; \backslash left(\backslash frac\{x+1\}\{2\}\backslash right)^\{n-s\}$

and for integer $n$

:

where $\backslash Gamma(z)$ is the gamma function.

In the special case that the four quantities $n$, $n+\backslash alpha$, $n+\backslash beta$, $n+\backslash alpha+\backslash beta$

are nonnegative integers, the Jacobi polynomial can be written as

{{NumBlk|:|$P\_n^\{(\backslash alpha,\backslash beta)\}(x)=(n+\backslash alpha)!\; (n+\backslash beta)!\; \backslash sum\_\{s=0\}^n\; \backslash frac\{1\}\{s!\; (n+\backslash alpha-s)!(\backslash beta+s)!(n-s)!\}\; \backslash left(\backslash frac\{x-1\}\{2\}\backslash right)^\{n-s\}\; \backslash left(\backslash frac\{x+1\}\{2\}\backslash right)^\{s\}.$|{{EquationRef|1}}}}

The sum extends over all integer values of $s$ for which the arguments of the factorials are nonnegative.

:$P\_0^\{(\backslash alpha,\backslash beta)\}(z)=\; 1,$

:$P\_1^\{(\backslash alpha,\backslash beta)\}(z)=\; (\backslash alpha+1)\; +\; (\backslash alpha+\backslash beta+2)\backslash frac\{z-1\}\{2\},$

:$P\_2^\{(\backslash alpha,\backslash beta)\}(z)=\; \backslash frac\{(\backslash alpha+1)(\backslash alpha+2)\}\{2\}$

+ (\alpha+2)(\alpha+\beta+3)\frac{z-1}{2}

+ \frac{(\alpha+\beta+3)(\alpha+\beta+4)}{2}\left(\frac{z-1}{2}\right)^2.

The Jacobi polynomials satisfy the orthogonality condition

:$\backslash int\_\{-1\}^1\; (1-x)^\{\backslash alpha\}\; (1+x)^\{\backslash beta\}\; P\_m^\{(\backslash alpha,\backslash beta)\}\; (x)P\_n^\{(\backslash alpha,\backslash beta)\}\; (x)\backslash ,dx\; =\backslash frac\{2^\{\backslash alpha+\backslash beta+1\}\}\{2n+\backslash alpha+\backslash beta+1\}\; \backslash frac\{\backslash Gamma(n+\backslash alpha+1)\backslash Gamma(n+\backslash beta+1)\}\{\backslash Gamma(n+\backslash alpha+\backslash beta+1)n!\}\; \backslash delta\_\{nm\},\; \backslash qquad\; \backslash alpha,\backslash \; \backslash beta\; >\; -1.$

As defined, they do not have unit norm with respect to the weight. This can be corrected by dividing by the square root of the right hand side of the equation above, when $n=m$.

Although it does not yield an orthonormal basis, an alternative normalization is sometimes preferred due to its simplicity:

:$P\_n^\{(\backslash alpha,\; \backslash beta)\}\; (1)\; =\; \{n+\backslash alpha\backslash choose\; n\}.$

The polynomials have the symmetry relation

:$P\_n^\{(\backslash alpha,\; \backslash beta)\}\; (-z)\; =\; (-1)^n\; P\_n^\{(\backslash beta,\; \backslash alpha)\}\; (z);$

thus the other terminal value is

:$P\_n^\{(\backslash alpha,\; \backslash beta)\}\; (-1)\; =\; (-1)^n\; \{\; n+\backslash beta\backslash choose\; n\}.$

The $k$th derivative of the explicit expression leads to

:$\backslash frac\{d^k\}\{dz^k\}\; P\_n^\{(\backslash alpha,\backslash beta)\}\; (z)\; =\; \backslash frac\{\backslash Gamma\; (\backslash alpha+\backslash beta+n+1+k)\}\{2^k\; \backslash Gamma\; (\backslash alpha+\backslash beta+n+1)\}\; P\_\{n-k\}^\{(\backslash alpha+k,\; \backslash beta+k)\}\; (z).$

The Jacobi polynomial $P\_n^\{(\backslash alpha,\backslash beta)\}$ is a solution of the second order linear homogeneous differential equation{{Pg|location=IV.2}}

:$\backslash left\; (1-x^2\; \backslash right\; )y\text{'}\text{'}\; +\; (\; \backslash beta-\backslash alpha\; -\; (\backslash alpha\; +\; \backslash beta\; +\; 2)x\; )y\text{'}\; +\; n(n+\backslash alpha+\backslash beta+1)\; y\; =\; 0.$

The recurrence relation for the Jacobi polynomials of fixed $\backslash alpha$, $\backslash beta$ is:{{Pg|location=IV.5}}

:$$

\begin{align}

&2n (n + \alpha + \beta) (2n + \alpha + \beta - 2) P_n^{(\alpha,\beta)}(z) \\

&\qquad= (2n+\alpha + \beta-1) \Big\{ (2n+\alpha + \beta)(2n+\alpha+\beta-2) z + \alpha^2 - \beta^2 \Big\} P_{n-1}^{(\alpha,\beta)}(z) - 2 (n+\alpha - 1) (n + \beta-1) (2n+\alpha + \beta) P_{n-2}^{(\alpha, \beta)}(z),

\end{align}

for $n=2,3,\backslash ldots$.

Writing for brevity $a:=n\; +\; \backslash alpha$, $b:=n\; +\; \backslash beta$ and $c:=a+b=2n\; +\; \backslash alpha+\; \backslash beta$, this becomes in terms of $a,b,c$

:$2n\; (c-n)(c-2)\; P\_n^\{(\backslash alpha,\backslash beta)\}(z)\; =(c-1)\; \backslash Big\backslash \{\; c(c-2)\; z\; +\; (a-b)(c-2n)\; \backslash Big\backslash \}\; P\_\{n-1\}^\{(\backslash alpha,\backslash beta)\}(z)-2\; (a-1)(b-1)\; c\backslash ;\; P\_\{n-2\}^\{(\backslash alpha,\; \backslash beta)\}(z).$

Since the Jacobi polynomials can be described in terms of the hypergeometric function, recurrences of the hypergeometric function give equivalent recurrences of the Jacobi polynomials. In particular, Gauss' contiguous relations correspond to the identities{{cite web|url=https://github.com/pec27/urdf | first1=P. E. | last1=Creasey | title = A Unitary BRDF for Surfaces with Gaussian Deviations}}{{Pg|location=Appx.B}}

:$$

\begin{align}

(z-1) \frac{d}{dz} P_n^{(\alpha,\beta)}(z)

& = \frac{1}{2} (z-1)(1+\alpha+\beta+n)P_{n-1}^{(\alpha+1,\beta+1)} \\

& = n P_n^{(\alpha,\beta)} - (\alpha+n) P_{n-1}^{(\alpha,\beta+1)} \\

& =(1+\alpha+\beta+n) \left( P_n^{(\alpha,\beta+1)} - P_{n}^{(\alpha,\beta)} \right) \\

& =(\alpha+n) P_n^{(\alpha-1,\beta+1)} - \alpha P_n^{(\alpha,\beta)} \\

& =\frac{2(n+1) P_{n+1}^{(\alpha,\beta-1)} - \left(z(1+\alpha+\beta+n)+\alpha+1+n-\beta \right) P_n^{(\alpha,\beta)}}{1+z} \\

& =\frac{(2\beta+n+nz) P_n^{(\alpha,\beta)} - 2(\beta+n) P_n^{(\alpha,\beta-1)}}{1+z} \\

& =\frac{1-z}{1+z} \left( \beta P_n^{(\alpha,\beta)} - (\beta+n) P_{n}^{(\alpha+1,\beta-1)} \right) \, .

\end{align}

The generating function of the Jacobi polynomials is given by

:$\backslash sum\_\{n=0\}^\backslash infty\; P\_n^\{(\backslash alpha,\backslash beta)\}(z)\; t^n\; =\; 2^\{\backslash alpha\; +\; \backslash beta\}\; R^\{-1\}\; (1\; -\; t\; +\; R)^\{-\backslash alpha\}\; (1\; +\; t\; +\; R)^\{-\backslash beta\},$

where

:$R\; =\; R(z,\; t)\; =\; \backslash left(1\; -\; 2zt\; +\; t^2\backslash right)^\{\backslash frac\{1\}\{2\}\}~,$

and the branch of square root is chosen so that $R(z,\; 0)\; =\; 1$.{{Pg|location=IV.4}}

For $x$ in the interior of $[-1,1]$, the asymptotics of $P\_n^\{(\backslash alpha,\backslash beta)\}$ for large $n$ is given by the Darboux formula{{Pg|location=VIII.2}}

:$P\_n^\{(\backslash alpha,\backslash beta)\}(\backslash cos\; \backslash theta)\; =\; n^\{-\backslash frac\{1\}\{2\}\}k(\backslash theta)\backslash cos\; (N\backslash theta\; +\; \backslash gamma)\; +\; O\; \backslash left\; (n^\{-\backslash frac\{3\}\{2\}\}\; \backslash right\; ),$

where

:$$

\begin{align}

k(\theta) &= \pi^{-\frac{1}{2}} \sin^{-\alpha-\frac{1}{2}} \tfrac{\theta}{2} \cos^{-\beta-\frac{1}{2}} \tfrac{\theta}{2},\\

N &= n + \tfrac{1}{2} (\alpha+\beta+1),\\

\gamma &= - \tfrac{\pi}{2} \left (\alpha + \tfrac{1}{2} \right ), \\

0 < \theta &< \pi

\end{align}

and the "$O$" term is uniform on the interval $[\backslash varepsilon,\backslash pi-\backslash varepsilon]$ for every $\backslash varepsilon>0$.

The asymptotics of the Jacobi polynomials near the points $\backslash pm\; 1$ is given by the Mehler–Heine formula

:$$

\begin{align}

\lim_{n \to \infty} n^{-\alpha}P_n^{(\alpha,\beta)}\left(\cos \left ( \tfrac{z}{n} \right ) \right) &= \left(\tfrac{z}{2}\right)^{-\alpha} J_\alpha(z)\\

\lim_{n \to \infty} n^{-\beta}P_n^{(\alpha,\beta)}\left(\cos \left (\pi - \tfrac{z}{n} \right) \right) &= \left(\tfrac{z}{2}\right)^{-\beta} J_\beta(z)

\end{align}

where the limits are uniform for $z$ in a bounded domain.

The asymptotics outside $[-1,1]$ is less explicit.

The expression ({{EquationNote|1}}) allows the expression of the Wigner d-matrix

$d^j\_\{m\text{'},m\}(\backslash phi)$ (for $0\backslash leq\; \backslash phi\backslash leq\; 4\backslash pi$)

in terms of Jacobi polynomials:{{cite book| last=Biedenharn| first=L.C.| last2=Louck| first2=J.D.|title=Angular Momentum in Quantum Physics|publisher=Addison-Wesley |location=Reading |year=1981}}

$d^j\_\{m\text{'}m\}(\backslash phi)\; =(-1)^\{\backslash frac\{m-m\text{'}-|m-m\text{'}|\}\{2\}\}\backslash left[\; \backslash frac\{(j+M)!(j-M)!\}\{(j+N)!(j-N)!\}\backslash right]^\{\backslash frac\{1\}\{2\}\}\; \backslash left(\backslash sin\backslash tfrac\{\backslash phi\}\{2\}\backslash right)^$

where $M\; =\; \backslash max(|m|,|m\text{'}|),\; N\; =\; \backslash min(|m|,|m\text{'}|)$.

• Askey–Gasper inequality
• Big q-Jacobi polynomials
• Continuous q-Jacobi polynomials
• Little q-Jacobi polynomials
• Pseudo Jacobi polynomials
• Jacobi process
• Gegenbauer polynomials
• Romanovski polynomials

• {{Citation | last1=Andrews | first1=George E. | last2=Askey | first2=Richard | last3=Roy | first3=Ranjan | title=Special functions | publisher=Cambridge University Press | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-62321-6 |id={{ISBN|978-0-521-78988-2}} | mr=1688958 | year=1999 | volume=71}}
• {{dlmf|id=18|title=Orthogonal Polynomials|first=Tom H. |last=Koornwinder|first2=Roderick S. C.|last2= Wong|first3=Roelof |last3=Koekoek||first4=René F. |last4=Swarttouw}}

• {{MathWorld|title=Jacobi Polynomial|urlname=JacobiPolynomial}}