Completeness of Legendre Polynomials

[ObsessiveMathsFreak]

I've recently been working with Legendre polynomials, particularly in the context of Spherical Harmonics. For the moment, it's enough to consider the regular L. polynomials which solve the differential equation

$$[(1-x^2) P_n']'+\lambda P=0$$

However, I've run into a problem. Why in the definition of spherical harmonics are only $$\lambda$$ of the form $$\lambda=n(n+1)$$ for integer n, accepted as valid solutions to the problem? In particular, with no boundary value specified, what restricts the eigenvalues to the problem here?

Note that my question is about the eigenvalues. I'm aware that the singular Legendre polynomials of the second kind (Q_n) should be rejected, but what stops a Legendre polynomial of the first kind with a non n(n+1) eigenvalue from being a solution?

I've read excerpts from Sturm-Liouville theory in an effort to find a solution to this problem, but mostly these texts simply repeat the same theorems that the eigenvalues are real, the solutions orthogonal, etc. There seems to be no method in the theory for proving that the eigenvalues $$n(n+1)$$ are the only eigenvalues.

So my question is this: Why are eigenvalues such as say $$\lambda=1$$ not permitted as solutions to the Legendre equation? Particularly in the absence of boundary values.

 

[AlephZero]

If $\lambda$ is not equal to $n(n+1)$, the regular (non-singular) solution to the equation is an infinite series, not a polynomial.

The set of polynomials of degrees 0, 1, 2, ... are obviously linearly independent, and they also have nice orthogonality properties. So there is no particular reason to want to use solutions for other values of $\lambda$ for anything. At least, Legendre didn't have any reason. If somebody else has found a use for them, I don't know about that.

I don't think there is anything more to this than simple pragmatism.

 

[ObsessiveMathsFreak]

I understand the pragmatic and aesthetic reasons fro excluding these solutions, and I follow the problem which arises in the infinite series for non integer n.

But where is the proof that the legendre polynomials are the only regular soultions to the differential equation?

 

[AlephZero]

But where is the proof that the legendre polynomials are the only regular soultions to the differential equation?

They aren't the only regular solutions. They are just the pragmatically useful ones.

The non-polynomial regular solutions are called "Legendre functions of the first kind". (The non-regular solutions are the second kind).

http://mathworld.wolfram.com/LegendreDifferentialEquation.html