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{{for|Jacobi polynomials of several variables|Heckman–Opdam polynomials}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
In [[mathematics]], '''Jacobi polynomials''' (occasionally called '''hypergeometric polynomials''') are a class of [[Classical orthogonal polynomials|classical]] [[orthogonal polynomials]]. They are orthogonal with respect to the weight


:<math> (1 - x)^\alpha (1+x)^\beta </math>
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on the interval [-1, 1]. The [[Gegenbauer polynomials]], and thus also the [[Legendre polynomials|Legendre]] and [[Chebyshev polynomials]], are special cases of the Jacobi polynomials.<ref name=sz>{{cite book | last1=Szegő | first1=Gábor | title=Orthogonal Polynomials | url=http://books.google.com/books?id=3hcW8HBh7gsC | publisher= American Mathematical Society | series=Colloquium Publications | isbn=978-0-8218-1023-1 | mr=0372517 | year=1939 | volume=XXIII|chapter=IV. Jacobi polynomials.}} The definition is in IV.1; the differential equation &ndash; in IV.2; Rodrigues' formula is in IV.3; the generating function is in IV.4; the recurrent relation is in IV.5.
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</ref>


The Jacobi polynomials were introduced by [[Carl Gustav Jacob Jacobi]].
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


==Definitions==
<!--'''PNG'''  (currently default in production)
===Via the hypergeometric function===
:<math forcemathmode="png">E=mc^2</math>


The Jacobi polynomials are defined via the [[hypergeometric function]] as follows<ref>{{Abramowitz_Stegun_ref|22|561}}</ref>:
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


:<math>P_n^{(\alpha,\beta)}(z)=\frac{(\alpha+1)_n}{n!}
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
\,_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\frac{1-z}{2}\right) ,</math>


where <math>(\alpha+1)_n</math> is [[Pochhammer symbol|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:
==Demos==


:<math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
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~.
</math>


===Rodrigues' formula===


An equivalent definition is given by [[Rodrigues' formula]]<ref name=sz/><ref>{{SpringerEOM|id=Jacobi_polynomials|author=P.K. Suetin}}</ref>:
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:<math>P_n^{(\alpha,\beta)} (z)
==Test pages ==
= \frac{(-1)^n}{2^n n!} (1-z)^{-\alpha} (1+z)^{-\beta}
\frac{d^n}{dz^n} \left\{ (1-z)^\alpha (1+z)^\beta (1 - z^2)^n \right\}~. </math>


===Alternate expression for real argument===
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For real ''x'' the Jacobi polynomial can alternatively be
*[[Inputtypes|Inputtypes (private Wikis only)]]
written as
*[[Url2Image|Url2Image (private Wikis only)]]
 
==Bug reporting==
:<math>P_n^{(\alpha,\beta)}(x)=
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
\sum_s
{n+\alpha\choose s}{n+\beta \choose n-s}
\left(\frac{x-1}{2}\right)^{n-s} \left(\frac{x+1}{2}\right)^{s}
</math>
 
where ''s'' ≥ 0 and ''n''-''s'' ≥ 0, and for integer ''n''
 
:<math>
{z\choose n} = \frac{\Gamma(z+1)}{\Gamma(n+1)\Gamma(z-n+1)},
</math>
 
and ''&Gamma;''(''z'') is the [[Gamma function]], using the convention that:
 
:<math>
{z\choose n} = 0 \quad\text{for}\quad n < 0.
</math>
 
In the special case that the four quantities
''n'', ''n''+''&alpha;'', ''n''+''&beta;'', and
''n''+''&alpha;''+''&beta;''  are nonnegative integers,
the Jacobi polynomial can be written as
 
{{NumBlk|:|<math>\begin{align}
&P_n^{(\alpha,\beta)}(x)=  (n+\alpha)! (n+\beta)! \\
&\qquad \times \sum_s
\left[s! (n+\alpha-s)!(\beta+s)!(n-s)!\right]^{-1} \left(\frac{x-1}{2}\right)^{n-s} \left(\frac{x+1}{2}\right)^{s}.
\end{align}
</math>|{{EquationRef|1}}}}
 
The sum extends over all integer values of ''s'' for which the arguments of the factorials are nonnegative.
 
==Basic properties==
===Orthogonality===
 
The Jacobi polynomials satisfy the orthogonality condition
 
:<math>\begin{align}
&\int_{-1}^1 (1-x)^{\alpha} (1+x)^{\beta}
P_m^{(\alpha,\beta)} (x)P_n^{(\alpha,\beta)} (x) \; dx \\
&\quad=
\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}
\end{align}
</math>
 
for ''&alpha;'' > -1 and ''&beta;'' > -1.
 
As defined, they are not orthonormal, the normalization being
 
:<math>P_n^{(\alpha, \beta)} (1) = {n+\alpha\choose n}.</math>
 
===Symmetry relation===
 
The polynomials have the symmetry relation
 
:<math>P_n^{(\alpha, \beta)} (-z) = (-1)^n P_n^{(\beta, \alpha)} (z);
</math>
 
thus the other terminal value is
:<math>P_n^{(\alpha, \beta)} (-1) = (-1)^n { n+\beta\choose n} .
</math>
 
===Derivatives===
 
The ''k''th derivative of the explicit expression leads to
 
:<math>
\frac{\mathrm d^k}{\mathrm d z^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) .
</math>
 
===Differential equation===
 
The Jacobi polynomial ''P''<sub>''n''</sub><sup>(''&alpha;'', ''&beta;'')</sup> is a solution of the second order [[linear homogeneous differential equation]]<ref name=sz/>
 
:<math>
(1-x^2)y'' + ( \beta-\alpha - (\alpha + \beta + 2)x )y'+ n(n+\alpha+\beta+1) y = 0.\,
</math>
 
===Recurrent relation===
 
The [[Orthogonal polynomials#Recurrence relations|recurrent relation]] for the Jacobi polynomials is<ref name=sz/>:
 
:<math>\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) \\
&\qquad\qquad - 2 (n+\alpha - 1) (n + \beta-1) (2n+\alpha + \beta)
    P_{n-2}^{(\alpha,\beta)}(z)~, \quad n = 2,3,\dots
\end{align}</math>
 
===Generating function===
 
The [[generating function]] of the Jacobi polynomials is given by
 
:<math> \sum_{n=0}^\infty P_n^{(\alpha,\beta)}(z) w^n
= 2^{\alpha + \beta} R^{-1} (1 - w + R)^{-\alpha} (1 + w + R)^{-\beta}~, </math>
 
where
 
:<math> R = R(z, w) = \big(1 - 2zw + w^2\big)^{1/2}~,  </math>
 
and the [[principal branch|branch]] of square root is chosen so that ''R''(''z'', 0) = 1.<ref name=sz/>
 
==Asymptotics of Jacobi polynomials==
 
For ''x'' in the interior of [-1, 1], the asymptotics of ''P''<sub>''n''</sub><sup>(''&alpha;'',''&beta;'')</sup> for large ''n'' is given by the Darboux formula<ref name=sz/>
 
:<math> P_n^{(\alpha,\beta)}(\cos \theta) = n^{-1/2} \cos (N\theta + \gamma) + O(n^{-3/2})~,</math>
 
where
 
:<math>\begin{align}
k(\theta) &= \pi^{-1/2} \sin^{-\alpha-1/2} \frac{\theta}{2} \cos^{-\beta-1/2} \frac{\theta}{2}~,\\
N &= n + \frac{\alpha+\beta+1}{2}~,\\
\gamma &= - (\alpha + \frac{1}{2}) \frac{\pi}{2}~,
\end{align} </math>
 
and the "''O''" term is uniform on the interval [''&epsilon;'', {{pi}}-''&epsilon;''] for every ''&epsilon;''>0.
 
The asymptotics of the Jacobi polynomials near the points ±1 is given by the [[Mehler&ndash;Heine formula]]
 
:<math>\begin{align}
\lim_{n \to \infty} n^{-\alpha}P_n^{\alpha,\beta}\left(\cos \frac{z}{n}\right)
&= \left(\frac{z}{2}\right)^{-\alpha} J_\alpha(z)~,\\
\lim_{n \to \infty} n^{-\beta}P_n^{\alpha,\beta}\left(\cos \left[ \pi - \frac{z}{n} \right] \right)
&= \left(\frac{z}{2}\right)^{-\beta} J_\beta(z)~,
\end{align}</math>
 
where the limits are uniform for ''z'' in a bounded [[Domain (mathematical analysis)|domain]].
 
The asymptotics outside [-1, 1] is less explicit.
 
==Applications==
===Wigner d-matrix===
The expression ({{EquationNote|1}}) allows the expression of the [[Wigner D-matrix#Wigner d-matrix|Wigner d-matrix]] ''d''<sup>''j''</sup><sub>''m''’,''m''</sub>(''&phi;'') (for 0 ≤ ''&phi;'' ≤ 4{{pi}}) in terms of Jacobi polynomials:<ref>{{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}}</ref>
 
:<math>\begin{align}
&d^j_{m'm}(\phi) =\left[
\frac{(j+m)!(j-m)!}{(j+m')!(j-m')!}\right]^{1/2} \\
&\qquad\times
\left(\sin\frac{\phi}{2}\right)^{m-m'}
\left(\cos\frac{\phi}{2}\right)^{m+m'}
P_{j-m}^{(m-m',m+m')}(\cos \phi).
\end{align}</math>
 
==See also==
 
*[[Askey–Gasper inequality]]
*[[Big q-Jacobi polynomials]]
*[[Continuous q-Jacobi polynomials]]
*[[Little q-Jacobi polynomials]]
*[[Pseudo Jacobi polynomials]]
*[[Jacobi process]]
 
==Notes==
<div class="references">
<references />
 
==Further reading==
 
*{{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; 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}}
 
==External links==
*{{MathWorld|title=Jacobi Polynomial|urlname=JacobiPolynomial}}
 
</div>
 
[[Category:Special hypergeometric functions]]
[[Category:Orthogonal polynomials]]
 
[[de:Jacobi-Polynom]]
[[fr:Polynôme de Jacobi]]
[[it:Polinomi di Jacobi]]
[[hu:Jacobi-polinomok]]
[[nl:Jacobi-polynoom]]
[[ru:Многочлены Якоби]]
[[fi:Jacobin polynomi]]
[[uk:Поліноми Якобі]]
[[vi:Đa thức Jacobi]]
[[zh:雅可比多项式]]

Latest revision as of 23:52, 15 September 2019

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