Jacobi polynomial

The Jacobi polynomials (after Carl Gustav Jacob Jacobi ), also hypergeometric polynomials, are a set of polynomial solutions of the Sturm-Liouville problem , which form a set of orthogonal polynomials on the interval with respect to the weight function with . They have the explicit form ${\displaystyle [-1,1]}$ ${\displaystyle (1-x)^{\alpha }(1+x)^{\beta }}$ ${\displaystyle \alpha ,\beta >-1}$

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

or with the help of the generalized hypergeometric function : ${\displaystyle {}_{2}F_{1}}$

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

Rodrigues formula

${\displaystyle P_{n}^{(\alpha ,\beta )}(x)={\frac {(-1)^{n}}{2^{n}n!}}(1-x)^{-\alpha }(1+x)^{-\beta }{\frac {d^{n}}{dx^{n}}}\left[(1-x)^{\alpha +n}(1+x)^{\beta +n}\right],~~~\alpha ,\beta >-1}$

Recursion formulas

The Jacobi polynomials can also be determined using a recursion formula.

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

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

${\displaystyle a_{n}^{1}P_{n+1}^{(\alpha ,\beta )}(x)=(a_{n}^{2}+a_{n}^{3}x)P_{n}^{(\alpha ,\beta )}(x)-a_{n}^{4}P_{n-1}^{(\alpha ,\beta )}(x)}$

with the constants:

${\displaystyle a_{n}^{1}=2(n+1)(n+\alpha +\beta +1)(2n+\alpha +\beta )}$

${\displaystyle a_{n}^{2}=(2n+\alpha +\beta +1)(\alpha ^{2}-\beta ^{2})}$

${\displaystyle a_{n}^{3}=(2n+\alpha +\beta )(2n+\alpha +\beta +1)(2n+\alpha +\beta +2)}$

${\displaystyle a_{n}^{4}=2(n+\alpha )(n+\beta )(2n+\alpha +\beta +2)}$

properties

The value for is ${\displaystyle x=1}$

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

The following symmetry relationship applies

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

which gives the value for : ${\displaystyle x=-1}$

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

They meet the orthogonality condition

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

Derivatives

The -th derivatives can be read from the explicit form . They result as: ${\displaystyle k}$

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

zeropoint

The eigenvalues ​​of the symmetric tridiagonal matrix

${\displaystyle {\begin{pmatrix}a_{0}&b_{1}&0&\dots &0\\b_{1}&a_{1}&b_{2}&\ddots &\vdots \\0&b_{2}&\ddots &\ddots &0\\\vdots &\ddots &\ddots &\ddots &b_{n-1}\\0&\dots &0&b_{n-1}&a_{n-1}\end{pmatrix}}}$

With

${\displaystyle a_{0}={\frac {\beta -\alpha }{2+\alpha +\beta }}}$

${\displaystyle a_{j}={\frac {(\beta -\alpha )(\alpha +\beta )}{(2j+\alpha +\beta )(2j+2+\alpha +\beta )}},~~~j=1,\dots ,n-1}$

${\displaystyle b_{j}={\sqrt {\frac {4j(j+\alpha )(j+\beta )(j+\alpha +\beta )}{(2j-1+\alpha +\beta )(2j+\alpha +\beta )^{2}(2j+1+\alpha +\beta )}}}}$

match the zeros of . The QR algorithm thus offers the possibility of calculating the zeros approximately. Furthermore, one can prove that they are simple and lie in the interval . ${\displaystyle P_{n}^{(\alpha ,\beta )}}$ ${\displaystyle (-1,1)}$

Asymptotic representation

With the help of the Landau symbols , the following formula can be set up:

${\displaystyle P_{n}^{(\alpha ,\beta )}(\cos \theta )={\frac {\cos \left(\left[n+(\alpha +\beta +1)/2\right]\theta -\left[2\alpha +1\right]\pi /4\right)}{{\sqrt {\pi n}}\left[\sin(\theta /2)\right]^{\alpha +1/2}\left[\cos(\theta /2)\right]^{\beta +1/2}}}+{\mathcal {O}}\left(n^{-3/2}\right),~~~0<\theta <\pi .}$

Generating function

For all true ${\displaystyle x\in \mathbb {R} ,z\in \mathbb {C} ,|z|<1}$

${\displaystyle \sum _{n=0}^{\infty }P_{n}^{(\alpha ,\beta )}(x)z^{n}=2^{\alpha +\beta }[f(x,z)]^{-1}[1-z+f(x,z)]^{-\alpha }[1+z+f(x,z)]^{-\beta },~~~f(x,z)={\sqrt {1-2xz+z^{2}}}.}$

The function

${\displaystyle z\mapsto 2^{\alpha +\beta }[f(x,z)]^{-1}[1-z+f(x,z)]^{-\alpha }[1+z+f(x,z)]^{-\beta }}$

is therefore called the generating function of the Jacobi polynomials.

Special cases

Some important polynomials can be viewed as special cases of the Jacobi polynomials:

• for : Legendre polynomials ${\displaystyle \alpha =\beta =0}$
• Gegenbauer polynomials
• Chebyshev polynomials of the first and second order
• the radial term of the Zernike polynomials

literature

• Eric W. Weisstein : Jacobi Polynomial . In: MathWorld (English).
• Sherwin Karniadakis: Spectral / hp Element Methods for CFD . 1st edition. Oxford University Press, New York 1999, ISBN 0-19-510226-6 .
• IS Gradshteyn, IM Ryzhik: Table of Integrals, Series, and Products . 5th edition. Academic Press Inc., Boston, San Diego, New York, London, Sydney, Tokyo, Toronto 1994, ISBN 0-12-294755-X .
• Peter Junghanns: EAGLE-GUIDE Orthogonal Polynomials . 1st edition. Books on Demand, Leipzig 2009, ISBN 3-937219-28-5 .

Individual evidence

1. ↑ Abramowitz, Stegun (1965): Formula 22.3.2 - also contains extensive additional information and evidence for the other formulas mentioned here