CMSC-203 Discrete Math: Special Problem 3 (spring 2000)
Difference Equations (50 bonus homework points)

Consider the following kth-order homogeneous linear difference equation (*) with constant coefficients:

T(n) + a_1 T(n-1) + a_2 T(n-2) + ... + a_k T(n-k) = 0
with initial conditions T(0) = t_0, T(1) = t_1, ..., T(k-1) = t_{k-1}
for some positive integer k, and
for some integral constants a_1, a_2, ..., a_k, and t_0, t_1, ..., t_{k-1}.

This difference equation defines a function T that maps the natural numbers to the integers. We seek a closed-form solution for T(n) expressed in terms of n.

Your theorem should give the general solution to Eq. (*). You should give necessary and sufficient conditions for the existence of this solution and rigorously prove that this solution is unique. Although you are encouraged to research prior work on this topic, you must express, organize, and explain your solution in your own words in a self-contained fashion appropriate for A students in CMSC-203 to understand. As much as reasonably possible, use methods and notations covered in CMSC-203. You may, however, appeal to fundamental ideas, terms, and facts in linear algebra without additional tutorial explanation. Include an annotated bibliography in your written solution of all sources that you consulted.

Your intended audience is an A student in CMSC-203 who is taking (or who has taken) linear algebra. Your written solution should be self-contained in the sense that such a student should be able to read and completely understand your solution without consulting any other source. The level of detail and explanation should be appropriate for the intended audience. Where appropriate you are welcome to introduce new concepts and notations; however, any new ideas and notations must be thoroughly explained, and there should be a compelling reason to introduce any such ideas or notations.

Note that your textbook neither addresses nor even mentions the issue of solution uniqueness. Also, your textbook discusses only special cases of Eq. (*) and leaves several claims unproven. In class, we learned a general technique for solving Eq. (*), but we did not rigorously state or prove the existence and uniqueness of the resulting general solution.

You are highly encouraged to prepare your written solution in LaTeX. Doing so will enable you to edit your work and to format mathematical expressions beautifully.

Your solution to this problem will earn up to 50 bonus points that will be added to your homework raw total. Handing in a quality solution to this bonus problem is an excellent way to demonstrate to the instructor your mastery of discrete math; such a demonstration of mastery will be given significant weight when final grades are assigned.

Checklist

  1. Have you stated and proven a theorem giving necessary and sufficient conditions for when Equation (*) has a solution?

  2. Have you stated and proven a theorem asserting that your solution to Equation (*) is unique?

  3. Do your theorems handle all possible cases for Equation (*), including the multiplicity greater than one, and complex root cases?

  4. If any of your answers to checklist questions 1-3 above is "no", then you need to do more work before handing in your solution.