orthopoly documentation (was: Howto simplify Integral with jacobi polynomials)

I looked this up once and here is what I claim to be true:


Jacobi polynomials are a class of orthogonal polynomials
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 $()_n$ is Pochhammer's symbol (for the rising factorial),
(Abramowitz and Stegun p561.) and thus have the explicit expression

\[
P_n^{(\alpha,\beta)} (z)
= \frac{\Gamma (\alpha+n+1)}{n!\Gamma (\alpha+\beta+n+1)}
\sum_{m=0}^n {n\choose m}
\frac{\Gamma (\alpha + \beta + n + m + 1)}{\Gamma (\alpha + m + 1)}
\left(\frac{z-1}{2}\right)^m .
\]


If this is in any way wrong, I would really like to know.




On Sat, Jul 12, 2008 at 7:45 AM, andre maute <andre.maute at gmx.de> wrote:
> On Friday 11 July 2008, Michel Talon wrote:
>> Raymond Toy (RT/EUS) wrote:
>> > andre maute wrote:
>> >> For the documentation, especially the orthopoly section,
>> >> Abramowitz & Stegun (A&S) is freely available
>> >> but Gradshteyn & Ryzhik and Merzbacher are not.
>> >>
>> >> Couldn't the documentation be a litlle bit more specific
>> >> at least for the nonfree citations?
>> >> Perhaps implementing the tables of A&S would help here also.
>> >
>> > I don't follow you.  What exactly are you suggesting here?  I, for one,
>> > would like it if the documentation gave some definition for each of the
>> > orthogonal polynomials.  The reference is nice, but, sometimes, I have
>> > neither net access nor book access.
>> >
>> > Ray
>>
>> For the spherical harmonics, there is a very reasonable account in
>> Wikipedia:
>> http://en.wikipedia.org/wiki/Spherical_harmonics
>> http://en.wikipedia.org/wiki/Associated_Legendre_functions
>
> I'm splitting the thread now.
>
> As stated in the wikipedia articles,
> there are different normalizations in use,
> Now could you tell me what kind of maxima uses,
> it is not stated in the documentation
>
> Andre
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