Jacobi polynomials

In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) ${\displaystyle P_{n}^{(\alpha ,\beta )}(x)}$ are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight ${\displaystyle (1-x)^{\alpha }(1+x)^{\beta }}$ on the interval ${\displaystyle [-1,1]}$. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials.^[1]

The Jacobi polynomials were introduced by Carl Gustav Jacob Jacobi.

Definitions

Via the hypergeometric function

The Jacobi polynomials are defined via the hypergeometric function as follows:^[2]

${\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {(\alpha +1)_{n}}{n!}}\,{}_{2}F_{1}\left(-n,1+\alpha +\beta +n;\alpha +1;{\tfrac {1}{2}}(1-z)\right),}$

where ${\displaystyle (\alpha +1)_{n}}$ 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:

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

Rodrigues' formula

An equivalent definition is given by Rodrigues' formula:^[1][3]

${\displaystyle P_{n}^{(\alpha ,\beta )}(z)={\frac {(-1)^{n}}{2^{n}n!}}(1-z)^{-\alpha }(1+z)^{-\beta }{\frac {d^{n}}{dz^{n}}}\left\{(1-z)^{\alpha }(1+z)^{\beta }\left(1-z^{2}\right)^{n}\right\}.}$

If ${\displaystyle \alpha =\beta =0}$, then it reduces to the Legendre polynomials:

${\displaystyle P_{n}(z)={\frac {1}{2^{n}n!}}{\frac {d^{n}}{dz^{n}}}(z^{2}-1)^{n}\;.}$

Alternate expression for real argument

For real ${\displaystyle x}$ the Jacobi polynomial can alternatively be written as

${\displaystyle P_{n}^{(\alpha ,\beta )}(x)=\sum _{s=0}^{n}{n+\alpha \choose n-s}{n+\beta \choose s}\left({\frac {x-1}{2}}\right)^{s}\left({\frac {x+1}{2}}\right)^{n-s}}$

and for integer ${\displaystyle n}$

${\displaystyle {z \choose n}={\begin{cases}{\frac {\Gamma (z+1)}{\Gamma (n+1)\Gamma (z-n+1)}}&n\geq 0\\0&n<0\end{cases}}}$

where ${\displaystyle \Gamma (z)}$ is the gamma function.

In the special case that the four quantities ${\displaystyle n}$, ${\displaystyle n+\alpha }$, ${\displaystyle n+\beta }$, ${\displaystyle n+\alpha +\beta }$ are nonnegative integers, the Jacobi polynomial can be written as

${\displaystyle P_{n}^{(\alpha ,\beta )}(x)=(n+\alpha )!(n+\beta )!\sum _{s=0}^{n}{\frac {1}{s!(n+\alpha -s)!(\beta +s)!(n-s)!}}\left({\frac {x-1}{2}}\right)^{n-s}\left({\frac {x+1}{2}}\right)^{s}.}$

(1)

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

Special cases

${\displaystyle P_{0}^{(\alpha ,\beta )}(z)=1,}$

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

${\displaystyle P_{2}^{(\alpha ,\beta )}(z)={\frac {(\alpha +1)(\alpha +2)}{2}}+(\alpha +2)(\alpha +\beta +3){\frac {z-1}{2}}+{\frac {(\alpha +\beta +3)(\alpha +\beta +4)}{2}}\left({\frac {z-1}{2}}\right)^{2}.}$

Basic properties

Orthogonality

The Jacobi polynomials satisfy the orthogonality condition

${\displaystyle \int _{-1}^{1}(1-x)^{\alpha }(1+x)^{\beta }P_{m}^{(\alpha ,\beta )}(x)P_{n}^{(\alpha ,\beta )}(x)\,dx={\frac {2^{\alpha +\beta +1}}{2n+\alpha +\beta +1}}{\frac {\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{\Gamma (n+\alpha +\beta +1)n!}}\delta _{nm},\qquad \alpha ,\ \beta >-1.}$

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 ${\displaystyle n=m}$.

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

${\displaystyle P_{n}^{(\alpha ,\beta )}(1)={n+\alpha \choose n}.}$

Symmetry relation

The polynomials have the symmetry relation

${\displaystyle P_{n}^{(\alpha ,\beta )}(-z)=(-1)^{n}P_{n}^{(\beta ,\alpha )}(z);}$

thus the other terminal value is

${\displaystyle P_{n}^{(\alpha ,\beta )}(-1)=(-1)^{n}{n+\beta \choose n}.}$

Derivatives

The ${\displaystyle k}$th derivative of the explicit expression leads to

${\displaystyle {\frac {d^{k}}{dz^{k}}}P_{n}^{(\alpha ,\beta )}(z)={\frac {\Gamma (\alpha +\beta +n+1+k)}{2^{k}\Gamma (\alpha +\beta +n+1)}}P_{n-k}^{(\alpha +k,\beta +k)}(z).}$

Differential equation

The Jacobi polynomial ${\displaystyle P_{n}^{(\alpha ,\beta )}}$ is a solution of the second order linear homogeneous differential equation^[1]

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

Recurrence relations

The recurrence relation for the Jacobi polynomials of fixed ${\displaystyle \alpha }$, ${\displaystyle \beta }$ is:^[1]

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

for ${\displaystyle n=2,3,\ldots }$. Writing for brevity ${\displaystyle a:=n+\alpha }$, ${\displaystyle b:=n+\beta }$ and ${\displaystyle c:=a+b=2n+\alpha +\beta }$, this becomes in terms of ${\displaystyle a,b,c}$

${\displaystyle 2n(c-n)(c-2)P_{n}^{(\alpha ,\beta )}(z)=(c-1){\Big \{}c(c-2)z+(a-b)(c-2n){\Big \}}P_{n-1}^{(\alpha ,\beta )}(z)-2(a-1)(b-1)c\;P_{n-2}^{(\alpha ,\beta )}(z).}$

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

${\displaystyle {\begin{aligned}(z-1){\frac {d}{dz}}P_{n}^{(\alpha ,\beta )}(z)&={\frac {1}{2}}(z-1)(1+\alpha +\beta +n)P_{n-1}^{(\alpha +1,\beta +1)}\\&=nP_{n}^{(\alpha ,\beta )}-(\alpha +n)P_{n-1}^{(\alpha ,\beta +1)}\\&=(1+\alpha +\beta +n)\left(P_{n}^{(\alpha ,\beta +1)}-P_{n}^{(\alpha ,\beta )}\right)\\&=(\alpha +n)P_{n}^{(\alpha -1,\beta +1)}-\alpha P_{n}^{(\alpha ,\beta )}\\&={\frac {2(n+1)P_{n+1}^{(\alpha ,\beta -1)}-\left(z(1+\alpha +\beta +n)+\alpha +1+n-\beta \right)P_{n}^{(\alpha ,\beta )}}{1+z}}\\&={\frac {(2\beta +n+nz)P_{n}^{(\alpha ,\beta )}-2(\beta +n)P_{n}^{(\alpha ,\beta -1)}}{1+z}}\\&={\frac {1-z}{1+z}}\left(\beta P_{n}^{(\alpha ,\beta )}-(\beta +n)P_{n}^{(\alpha +1,\beta -1)}\right)\,.\end{aligned}}}$

Generating function

The generating function of the Jacobi polynomials is given by

${\displaystyle \sum _{n=0}^{\infty }P_{n}^{(\alpha ,\beta )}(z)t^{n}=2^{\alpha +\beta }R^{-1}(1-t+R)^{-\alpha }(1+t+R)^{-\beta },}$

where

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

and the branch of square root is chosen so that ${\displaystyle R(z,0)=1}$.^[1]

Asymptotics of Jacobi polynomials

For ${\displaystyle x}$ in the interior of ${\displaystyle [-1,1]}$, the asymptotics of ${\displaystyle P_{n}^{(\alpha ,\beta )}}$ for large ${\displaystyle n}$ is given by the Darboux formula^[1]

${\displaystyle P_{n}^{(\alpha ,\beta )}(\cos \theta )=n^{-{\frac {1}{2}}}k(\theta )\cos(N\theta +\gamma )+O\left(n^{-{\frac {3}{2}}}\right),}$

where

${\displaystyle {\begin{aligned}k(\theta )&=\pi ^{-{\frac {1}{2}}}\sin ^{-\alpha -{\frac {1}{2}}}{\tfrac {\theta }{2}}\cos ^{-\beta -{\frac {1}{2}}}{\tfrac {\theta }{2}},\\N&=n+{\tfrac {1}{2}}(\alpha +\beta +1),\\\gamma &=-{\tfrac {\pi }{2}}\left(\alpha +{\tfrac {1}{2}}\right),\\0<\theta &<\pi \end{aligned}}}$

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

The asymptotics of the Jacobi polynomials near the points ${\displaystyle \pm 1}$ is given by the Mehler–Heine formula

${\displaystyle {\begin{aligned}\lim _{n\to \infty }n^{-\alpha }P_{n}^{(\alpha ,\beta )}\left(\cos \left({\tfrac {z}{n}}\right)\right)&=\left({\tfrac {z}{2}}\right)^{-\alpha }J_{\alpha }(z)\\\lim _{n\to \infty }n^{-\beta }P_{n}^{(\alpha ,\beta )}\left(\cos \left(\pi -{\tfrac {z}{n}}\right)\right)&=\left({\tfrac {z}{2}}\right)^{-\beta }J_{\beta }(z)\end{aligned}}}$

where the limits are uniform for ${\displaystyle z}$ in a bounded domain.

The asymptotics outside ${\displaystyle [-1,1]}$ is less explicit.

Applications

Wigner d-matrix

The expression (1) allows the expression of the Wigner d-matrix ${\displaystyle d_{m',m}^{j}(\phi )}$ (for ${\displaystyle 0\leq \phi \leq 4\pi }$) in terms of Jacobi polynomials:^[4]

${\displaystyle d_{m'm}^{j}(\phi )=\left[{\frac {(j+m)!(j-m)!}{(j+m')!(j-m')!}}\right]^{\frac {1}{2}}\left(\sin {\tfrac {\phi }{2}}\right)^{m-m'}\left(\cos {\tfrac {\phi }{2}}\right)^{m+m'}P_{j-m}^{(m-m',m+m')}(\cos \phi ).}$

See also

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

Notes