Linear difference equation with vanishing inhomogeneity

[feuerstein]

Hi,

I am new to this forum and apologize for bothering you with my question on recurrence/difference equations. Unfortunately I was unable to find an answer in textbooks etc., so I would really appreciate a short answer or a reference to relevant literature. The problem is as follows:

Say we have an order-d linear recurrence/difference equation:

a_n = c_1 a_{n-1} + c_2 a_{n-2} + ... + c_d a_{n-d}

We are interested in stability, i.e. whether the sequence a_n has a
limit as n goes to infinity. According to standard theorems convergence
occurs if and only if all roots of the corresponding characteristic
equation have absolute value less than 1.

My problem is that I do not quite have the above recurrence, but a
recurrence that "converges" to the above. More precisely, my recurrence is

b_n = c_1 b_{n-1} + c_2 b_{n-2} + ... + c_d b_{n-d} + \alpha(n),

where \alpha(n) is a sequence that converges to 0.

The questions are:

- Is b_n stable under the same conditions as for a_n?
- Are the limits of a_n and b_n the same (if any)?



Feuerstein

 

[jvicens]

Hello Feuerstein,

First of all, welcome to the forum! It's great to have new members who are interested in learning and discussing scientific topics.

To answer your questions, yes, the stability conditions for b_n are the same as for a_n. As long as all the roots of the corresponding characteristic equation have absolute value less than 1, b_n will also converge to a limit as n goes to infinity. This is because, as you mentioned, \alpha(n) converges to 0 and therefore has a negligible effect on the overall behavior of the recurrence.

As for the limits of a_n and b_n, they may not necessarily be the same. In some cases, they may converge to different limits depending on the initial values of the sequences and the values of the coefficients c_i. However, if the recurrence is well-behaved and the roots of the characteristic equation are distinct, then the limits of a_n and b_n should be the same.

I hope this helps answer your questions. If you need further clarification or would like a reference to relevant literature, please don't hesitate to ask. Happy exploring!