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I have a question; help is welcome.
Let sn be a linear, non-homogeneous recurrence sequence, and let hn be a corresponding homogeneous sequence (with initial values to be determined).
As it turns out, the offset between the two (sn - hn) is given by the steady state value of sn, if the initial values of hn are offset from those of sn by the same amount. Precisely what is the reason for that?
This steady state value bears no relation to the initial values of the sequence sn; more properly, it should be called the steady state value of the recurrence rule. I believe it is clear that a linear recurrence rule will have exactly one steady state value, neither none nor multiple (as the steady state is given by the root of a first-degree polynomial). And the steady state of hn is, of course, 0 (the root of a first-degree polynomial through the origin -- d'oh!). Therefore, if the desired offset does not depend on the initial values of sn (the offset was hand-set on the initial values, but who says it will stay that way further on in the sequences?), then the difference of the two steady states (thus the steady state of sn, as the one of hn is zero) should do. But why is the part in bold true?
Thanks--
Let sn be a linear, non-homogeneous recurrence sequence, and let hn be a corresponding homogeneous sequence (with initial values to be determined).
As it turns out, the offset between the two (sn - hn) is given by the steady state value of sn, if the initial values of hn are offset from those of sn by the same amount. Precisely what is the reason for that?
This steady state value bears no relation to the initial values of the sequence sn; more properly, it should be called the steady state value of the recurrence rule. I believe it is clear that a linear recurrence rule will have exactly one steady state value, neither none nor multiple (as the steady state is given by the root of a first-degree polynomial). And the steady state of hn is, of course, 0 (the root of a first-degree polynomial through the origin -- d'oh!). Therefore, if the desired offset does not depend on the initial values of sn (the offset was hand-set on the initial values, but who says it will stay that way further on in the sequences?), then the difference of the two steady states (thus the steady state of sn, as the one of hn is zero) should do. But why is the part in bold true?
Thanks--