[Maxima-bugs] [ maxima-Bugs-2862197 ] assoc_legendre_p(n, -1, x) wrong sign

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Bugs item #2862197, was opened at 2009-09-19 17:15
Message generated for change (Comment added) made by crategus
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Category: Lisp Core - Polynomials
Group: None
Status: Open
Resolution: None
Priority: 5
Private: No
Submitted By: Dieter Kaiser (crategus)
Assigned to: Nobody/Anonymous (nobody)
Summary: assoc_legendre_p(n,-1,x) wrong sign

Initial Comment:
The sign of assoc_legendre_p(n,-1,x) is wrong. We get:

(%i32) assoc_legendre_p(1,-1,x);
(%o32) -sqrt(1-x^2)/2

(%i33) assoc_legendre_p(2,-1,x);
(%o33) -x*sqrt(1-x^2)/2

(%i36) factor(ratsimp(assoc_legendre_p(3,-1,x)));
(%o36) -sqrt(1-x^2)*(5*x^2-1)/8

(%i37) factor(ratsimp(assoc_legendre_p(4,-1,x)));
(%o37) -x*sqrt(1-x^2)*(7*x^2-3)/8

In all cases above we obtain the expected answer when we multiply the result of Maxima with -1.
As a reference I have taken the results from wolfram alpha.

This bug is related to the problem that 

   specint(exp(-s*t)*t^n*bessel_j(1,t),t)

does not work for a general power n.

Dieter Kaiser


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>Comment By: Dieter Kaiser (crategus)
Date: 2009-09-19 18:36

Message:
I had a look into the code. We do the following transformation for a
negative second parameter m:

   assoc_legendre_p(n,-m,x) -> factorial(n+m)/factorial(n-m) *
asscoc_legendre_p(n,m,x)

That is the simplified formula A&S 8.2.5 for m an integer.

Wolfram functions gives an additional factor (-1)^m. I think wolfram
function is correct. With this factor we would get the expected results for
the Laplace transform of t^n*bessel_j(1,t) too.

Furthermore we would get the correct result for the Laplace transform of
t^u*bessel_j(v,z). The expected answer of example 73 in rtest14.mac is not
correct. We can see it when we insert specific values in the result.

Dieter Kaiser


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