Classical Orthogonal Polynomials - Differential Equation

Differential Equation

The classical orthogonal polynomials arise from a differential equation of the form

where Q is a given quadratic (at most) polynomial, and L is a given linear polynomial. The function f, and the constant λ, are to be found.

(Note that it makes sense for such an equation to have a polynomial solution.

Each term in the equation is a polynomial, and the degrees are consistent.)

This is a Sturm-Liouville type of equation. Such equations generally have singularities in their solution functions f except for particular values of λ. They can be thought of an eigenvector/eigenvalue problems: Letting D be the differential operator, and changing the sign of λ, the problem is to find the eigenvectors (eigenfunctions) f, and the corresponding eigenvalues λ, such that f does not have singularities and D(f) = λf.

The solutions of this differential equation have singularities unless λ takes on specific values. There is a series of numbers λ₀, λ₁, λ₂, ... that lead to a series of polynomial solutions P₀, P₁, P₂, ... if one of the following sets of conditions are met:

1. Q is actually quadratic, L is linear, Q has two distinct real roots, the root of L lies strictly between the roots of Q, and the leading terms of Q and L have the same sign.
2. Q is not actually quadratic, but is linear, L is linear, the roots of Q and L are different, and the leading terms of Q and L have the same sign if the root of L is less than the root of Q, or vice-versa.
3. Q is just a nonzero constant, L is linear, and the leading term of L has the opposite sign of Q.

These three cases lead to the Jacobi-like, Laguerre-like, and Hermite-like polynomials, respectively.

In each of these three cases, we have the following:

• The solutions are a series of polynomials P₀, P₁, P₂, ..., each P_n having degree n, and corresponding to a number λ_n.
• The interval of orthogonality is bounded by whatever roots Q has.
• The root of L is inside the interval of orthogonality.
• Letting, the polynomials are orthogonal under the weight function
• W(x) has no zeros or infinities inside the interval, though it may have zeros or infinities at the end points.
• W(x) gives a finite inner product to any polynomials.
• W(x) can be made to be greater than 0 in the interval. (Negate the entire differential equation if necessary so that Q(x) > 0 inside the interval.)

Because of the constant of integration, the quantity R(x) is determined only up to an arbitrary positive multiplicative constant. It will be used only in homogeneous differential equations (where this doesn't matter) and in the definition of the weight function (which can also be indeterminate.) The tables below will give the "official" values of R(x) and W(x).