Talk:Linear recurrence with constant coefficients

The article does not really clarify whether the following sequence: ${\displaystyle 1,0,0,0,0,0,...}${\displaystyle 1,0,0,0,0,0,...} is a valid solution? According to the definition, it might be because it satisfies the following linear difference equation: ${\displaystyle y_{0}=1,y_{1}=0,{\text{ and }}\forall t\in \mathbb {N} ,y_{t+2}=y_{t+1}+0y_{t}}${\displaystyle y_{0}=1,y_{1}=0,{\text{ and }}\forall t\in \mathbb {N} ,y_{t+2}=y_{t+1}+0y_{t}}. This is consistent with the article saying that the equation holds for "the values of the elements of a sequence" (emphasis mine). Note that it is not stated that the last coefficient must be nonzero, i.e. ${\displaystyle a_{n}\neq 0}${\displaystyle a_{n}\neq 0}. The page Constant-recursive sequence appears to allow eventually-periodic sequences.

However, it would not be allowed if the equation must hold for all ${\displaystyle t\in \mathbb {Z} }${\displaystyle t\in \mathbb {Z} }, rather than just ${\displaystyle t\in \mathbb {N} }${\displaystyle t\in \mathbb {N} }. And my impression is that most of the results in the article (e.g., closed formula in powers of the characteristic roots) would not hold for eventually-periodic sequences. Caleb Stanford (talk) 18:51, 7 November 2021 (UTC) [reply ]

I completed an initial merge, keeping unique content while removing duplicated content between the two articles. However, due to notation differences and narrative choices most of the material needs major rewriting to fit in the new article, and this may be too great a task for me to personally take on. For now, I put Template:under construction tags. The next step would be a pass to unify the notation. Caleb Stanford (talk) 20:46, 3 January 2022 (UTC) [reply ]

Hi, I wonder what is the scope difference between this article and Constant-recursive sequence? It seems they're roughly about the same concept. adamant.pwn — contrib/talk 17:26, 20 February 2022 (UTC) [reply ]

@Adamant.pwn: The short answer is that constant-recursive sequence is about sequences, while this page is about solving recurrence relations. In terms of actual material there is actually not much overlap between the pages -- I've tried to separate it out so that constant-recursive sequence does not discuss how to solve recurrences at all. I had at some point tried to merge the two and it did not work out well. Caleb Stanford (talk) 21:41, 21 February 2022 (UTC) [reply ]

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