Jacobi polynomials

In mathematics, Jacobi polynomials are a class of orthogonal polynomials. They are obtained from hypergeometric series in cases where the series is in fact finite:

:

where $(alpha+1)\_n$ is Pochhammer's symbol (for the rising factorial), (Abramowitz & Stegun [http://www.math.sfu.ca/~cbm/aands/page_561.htm p561] .) and thus have the explicit expression

:

from which the terminal value follows

:Here for integer $n,$:$\{zchoose\; n\}\; =\; frac\{Gamma(z+1)\}\{Gamma(n+1)Gamma(z-n+1)\},$and $Gamma(z),$ is the usual Gamma function, which has the property$1/Gamma(n+1)\; =\; 0,$ for $n=-1,-2,dots,$. Thus,:

They satisfy the orthogonality condition:for $alpha>-1$ and .

The polynomials have the symmetry relation ; thus the other terminal value is

:

For real $x$ the Jacobi polynomial can alternatively bewritten as:where $s\; ge\; 0\; ,$ and $n-s\; ge\; 0\; ,$.In the special case that the four quantities$n$, $n+alpha$, , and are nonnegative integers,the Jacobi polynomial can be written as:The sum on $s,$ extends over all integer values for which the arguments of the factorials are nonnegative.

This form allows the expression of the Wigner d-matrix $d^j\_\{m\text{'}\; m\}(phi);$ ($0le\; phile\; 4pi$) in termsof Jacobi polynomials [L. C. Biedenharn and J. D. Louck, "Angular Momentum in Quantum Physics", Addison-Wesley, Reading, (1981)] :$d^j\_\{m\text{'}m\}(phi)\; =left\; [frac\{(j+m)!(j-m)!\}\{(j+m\text{'})!(j-m\text{'})!\}\; ight]\; ^\{1/2\}left(sinfrac\{phi\}\{2\}\; ight)^\{m-m\text{'}\}left(cosfrac\{phi\}\{2\}\; ight)^\{m+m\text{'}\}P\_\{j-m\}^\{(m-m\text{'},m+m\text{'})\}(cos\; phi).$

Derivatives

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

:

Differential equation

Jacobi polynomials are solution of

:

References

Cited referencesGeneral references
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*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 | id=MathSciNet | id = 1688958 | year=1999 | volume=71