Talk:Constant-recursive sequence: Difference between revisions

Revision as of 01:33, 8 November 2021

Revision as of 01:35, 8 November 2021

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:: Good point. I checked The Concrete Tetrahedron (by Kauers and Paule), and they do require <math>c_d \ne 0</math>. This is artificial from the perspective of generating series because then the characterization as rational series needs an extra requirement on the degree of the numerator. But on the other hand it does fix the characterization as exponential polynomials. [[User:Eric Rowland|Eric Rowland]] ([[User talk:Eric Rowland|talk]]) 00:47, 8 November 2021 (UTC)

::: Thanks for looking into another reference! My preference would then be to have <math>c_d \ne 0</math>. <s>Three Two</s> One other justificationsjustification<s>s</s> for that criterion: (i) it allows the sequence to be uniquely extended in the negative direction as well, <s>(ii) it's implied by the "alternate definition" under "Definition" in the article, namely "the set of sequences <math>\{s(n + r)_{n \geq 0} : r \geq 0\}</math> is contained in a [[vector space]] whose dimension is finite."</s> For <math>1, 0, 0, 0, ...</math> we get the set of all finite-support sequences, which is infinite-dimensional.</s> [[User:Caleb Stanford|Caleb Stanford]] ([[User talk:Caleb Stanford|talk]]) 01:06, 8 November 2021 (UTC)

:::: I'm fine with requiring <math>c_d \ne 0</math>, if you want to make the change. I agree with your point (i). For your point (ii), if <math>s(n)_{n \geq 0}</math> is the sequence <math>1, 0, 0, 0, ...</math>, then <math>s(n + 1)_{n \geq 0}</math> and all higher shifts are the zero sequence, right? In that case <math>\{s(n + r)_{n \geq 0} : r \geq 0\}</math> is contained in the <math>1</math>-dimensional space of sequences of the form <math>a, 0, 0, 0, ...</math>. [[User:Eric Rowland|Eric Rowland]] ([[User talk:Eric Rowland|talk]]) 01:17, 8 November 2021 (UTC)