# [Maxima] Howto simplify Integral with jacobi polynomials

andre maute andre.maute at gmx.de
Thu Jul 10 12:22:45 CDT 2008

```On Thursday 10 July 2008, 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?
One point for a better documentation could also be to
have more functionality,

e.g.

1. L2-Norm (h_n in A&S),
2. leading coefficient (k_n in A&S)
3. rodriguez coeffcient (a_n in A&S)
4. coefficients in a special linearcombination (d_n, c_m in A&S)
these are tabulated on page 775
5. with the assumptions and declarations from my previous post i get
-----------------------------------------------------------------
(%i12) jacobi_p(k,a,b,-1)
(%o12) pochhammer(a+1,k)*('sum(pochhammer(-k,i)*pochhammer(k+b+a+1,i)
*pochhammer(a+1,i)^-1*i!^-1
*1^i,i,0,k))
------------------------------------------------------------------
there is a really nicer form for this one see page 777

6. the coefficients of the second order ODE are missing (page 781 in A&S)
7. the discrete orthogonal polynomials from A&S are missing

> I, for one, would like it if the documentation
> gave some definition for each of the orthogonal polynomials.
Exactly.
E.g.
If I want to verify a hand calculation with maxima, but don't know
what maxima uses as definitions for the orthogonal polynomial

One could inspect the lisp code for orthopoly
but there a hypergeometric identity is used,
which I doubt is useful for the average user.
I also doubt that the average user is used to lisp.

But if you have the above coefficients you could guess
what definition maxima uses.

> The reference is nice, but, sometimes, I have
> neither net access nor book access.
There is a single reference to Merzbacher,
which refers to the definition of the spherical harmonics.
nothing more.