Jacobi polynomials

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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:

P_n^{(\alpha,\beta)}(z)=\frac{(\alpha+1)_n}{n!} \,_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\frac{1-z}{2}\right) ,

where (α + 1)n is Pochhammer's symbol (for the rising factorial), (Abramowitz & Stegun p561.) and thus have the explicit expression

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 ,

from which the terminal value follows

P_n^{(\alpha, \beta)} (1) = {n+\alpha\choose n} .

Here for integer n\,

{z\choose 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,

{z\choose n} = 0 \quad\hbox{for}\quad n < 0.

The polynomials have the symmetry relation P_n^{(\alpha, \beta)} (-z) = (-1)^n P_n^{(\beta, \alpha)} (z) ; thus the other terminal value is

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

For real x the Jacobi polynomial can alternatively be written as

P_n^{(\alpha,\beta)}(x)= \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}

where s \ge 0 \, and n-s \ge 0 \, . In the special case that the four quantities n, n + α, n + β, and n + α + β are nonnegative integers, the Jacobi polynomial can be written as

P_n^{(\alpha,\beta)}(x)= (n+\alpha)! (n+\beta)! \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}.

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' m}(\phi)\; (0\le \phi\le 4\pi) in terms of Jacobi polynomials [1]

d^j_{m'm}(\phi) =\left[ \frac{(j+m)!(j-m)!}{(j+m')!(j-m')!}\right]^{1/2} \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).

[edit] Derivatives

The k-th derivative of the explicit expression leads to

\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) .

[edit] Differential equation

Jacobi polynomials P_n^{(\alpha,\beta)}are solution of

(1 − x2)y'' + (β − α − (α + β + 2)x)y' + n(n + α + β + 1)y = 0

[edit] References

Cited references

  1. ^ L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics, Addison-Wesley, Reading, (1981)

General references