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] References

Cited references

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

General references

Milton Abramowitz and Irene A. Stegun, eds. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover. ISBN 0-486-61272-4. (See chapter 22)