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:
:
,_2F_1left(-n,1+lpha+eta+n;lpha+1;rac{1-z}{2}
ight) ,
where is Pochhammer's symbol (for the rising factorial), (Abramowitz & Stegun p561.) and thus have the explicit expression
:$$
P_n^{(lpha,eta)} (z) =
rac{Gamma (lpha+n+1)}{n!Gamma (lpha+eta+n+1)}
sum_{m=0}^n {nchoose m}
rac{Gamma (lpha + eta + n + m + 1)}{Gamma (lpha + m + 1)} left(rac{z-1}{2}
ight)^m ,

from which the terminal value follows
:
Here for integer $n,$
:$$
{zchoose n} = rac{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,
:$$
{zchoose n} = 0 quadhbox{for}quad n < 0.

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

For real $x$ the Jacobi polynomial can alternatively be
written as
:
sum_s
{n+lphachoose s}{n+eta choose n-s}
left(rac{x-1}{2}
ight)^{n-s} left(rac{x+1}{2}
ight)^{s}

where $s\; ge\; 0\; ,$ and $n-s\; ge\; 0\; ,$.
In the special case that the four quantities
$n$, , , and
are nonnegative integers,
the Jacobi polynomial can be written as
:
sum_s
left[s! (n+lpha-s)!(eta+s)!(n-s)!
ight]^{-1}
left(rac{x-1}{2}
ight)^{n-s} left(rac{x+1}{2}
ight)^{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 $d^j\_\{m\text{'}\; m\}(phi);$
($0le\; phile\; 4pi$) in terms
of Jacobi polynomials ^[1]
:$$
d^j_{m'm}(phi) =left[
rac{(j+m)!(j-m)!}{(j+m')!(j-m')!}
ight]^{1/2}
left(sinrac{phi}{2}
ight)^{m-m'}
left(cosrac{phi}{2}
ight)^{m+m'}
P_{j-m}^{(m-m',m+m')}(cos phi).

Contents

Derivatives

References

Derivatives

The $k$-th derivative of the explicit expression leads to
:$$
rac{mathrm d^k}{mathrm d z^k}
P_n^{(lpha,eta)} (z) =
rac{Gamma (lpha+eta+n+1+k)}{2^k Gamma (lpha+eta+n+1)}
P_{n-k}^{(lpha+k, eta+k)} (z) .

References

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

'General references'

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