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We investigate the Hill differential equation where A(t), B(t), and D(t) are trigonometric polynomials. We are interested in solutions that are even or odd, and have period π or semi-period π . The above equation with one of the above conditions constitutes a regular Sturm-Liouville eigenvalue problem. We investigate the representation of the four Sturm-Liouville operators by infinite banded matrices.

The first known appearance of the Ince equation,

is in Whittaker’s paper ( [

One of the important features of the Ince equation is that the corresponding Ince differential operator when applied to Fourier series can be represented by an infinite tridiagonal matrix. It is this part of the theory that makes the Ince equation particularly interesting. For instance, the coexistence problem which has no simple solution for the general Hill equation has a complete solution for the Ince equation (see [

When studying the Ince equation, it became apparent that many of its properties carry over to a more general class of equations “the generalized Ince equation”. These linear second order differential equations describe important physical phenomena which exhibit a pronounced oscillatory character; behavior of pendulum-like systems, vibrations, resonances and wave propagation are all phenomena of this type in classical mechanics, (see for example [

We consider the Hill differential equation

where

Here

The real number

otherwise solutions

The polynomials

will play an important role in the analysis of (2.1). For ease of notation we also introduce the polynomials

Equation (2.1) is a natural generalization to the original Ince equation

Ince’s equation by itself includes some important particular cases, if we choose for example

with associated pzlynomial

If we choose

with associated polynomial

Equation (2.1) can be brought to algebraic form by applying the transformation

Equation (2.1) is an even Hill equation with period

Lemma 3.1. Let

Equation (2.1) can be written in the self adjoint form

where

Note that

Proof. Let

which is equivalent to

Noting that

and

we see that

Therefore, (3.8) can be written as

Since

In the case of Ince’s Equation (2.4), we have the following formula for the function

When

Equation (2.1) with one of the boundary conditions in lemma 3.1 is a regular Sturm-Liouville problem. From the theory of Sturm-Liouville ordinary differential equations it is known that such an eigenvalue problem has a sequence of eigenvalues that converge to infinity. These eigen values are denoted by

Theorem 3.2. The generalized Ince equation admits a nontrivial even solution with period

Example 3.3. To gain some understanding about the notation we consider the almost trivial completely solvable example, the so called Cauchy boundary value problem

subject to the boundary conditions of lemma 3.1. We have the following for the eigenvalues

1) Even with period

2) Even with semi-period

3) Odd with semi-period

4) Odd with semi-period

The formal adjoint of the generalized Ince equation is

By introducing the functions

we note that the adjoint of (2.1) has the same form and can be written in the following form:

Lemma 3.4. If

Proof. We Know that

and

For ease of notation, let

then

Substituting for

From lemma 3.4 we know that if

Theorem 3.5. We have for

From Sturm-Liouville theory we obtain the following statement on the distribution of eigenvalues.

Theorem 3.6. The eigenvalues of the generalized Ince equation satisfy the inequalities

The theory of Hill equation [

Theorem 3.7. If

for some positive integer

By theorem 3.2, the generalized Ince’s equation with

The generalized Ince’s equation with

The generalized Ince equation with

The generalized Ince equation with

From Sturm-Liouville theory ( [

Theorem 4.1. Each of the function systems

is orthogonal over

Moreover, each of the previous system is complete over

Using the transformations that led to Theorem 3.5, we obtain the following result.

Theorem 4.2. We have

where

with

The adopted normalization of Ince functions is easily expressible in terms of the Fourier coefficients of Ince functions and so is well suited for numerical computations [

Of course, once the generalized Ince functions

If we square both sides of (4.13) and (4.14) and integrate, we find that

If

Once we know

The integrals on the right-hand sides of (4.19) and (4.20) are easy to calculate once we know the Fourier series of Ince functions.

In this section we introduce four linear operators associated with Equation (2.1), and represent them by banded matrices of width

Let

By restricting functions to

We consider the differential operator

The domain

on the same domain

defines a bijective linear map

Let

can be represented in the following way,

where

Now consider the operator

defines a bijective linear map

Let

where

Now consider the operator

tions with semi-period

defines a bijective linear map

Let

where

and

Finally, consider the operator

defines a bijective linear map

Let

where

Example 5.1. For the Whittaker-Hill Equation (2.7) in the following form [

the function

Hilbert spaces

The generalized Ince functions admit the following Fourier series expansions

We did not indicate the dependence of the Fourier coefficients on

Using relations (4.13) and (4.14), we can represent the generalized functions in a different way

where

Therefore, we can write

where

and the Fourier coefficients

A generalized Ince function is called a generalized Ince polynomial of the first kind if its Fourier series (6.1), (6.2), (6.3), or (6.4) terminate. It is called a generalized Ince polynomial of the second kind if its expansion (6.11), (6.12), (6.13), or (6.14) terminate. If they exist, these generalized Ince polynomials and their corresponding eigenvalues can be computed from the finite subsections of the matrices

Example 6.1. Consider the equation

one can check that if we set

eigenvalue