Timeline for "Approximating" linear recursion with homogenous polynomial coefficients by linear recursion with constant coefficients
Current License: CC BY-SA 4.0
12 events
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| Oct 23, 2020 at 5:14 | vote | accept | asrxiiviii | ||
| Oct 23, 2020 at 3:49 | comment | added | Iosif Pinelis | @asrxiiviii : Your conjecture $A_n\asymp C_n$ has now been rigorously disproved, with the help of a general result found in the literature. | |
| Oct 23, 2020 at 3:44 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Oct 21, 2020 at 21:08 | comment | added | Iosif Pinelis | @asrxiiviii : I have added the mentioned plotting. I think I could prove rigorously that in your example $A_n\not\asymp C_n$, but am afraid that the effort to prove such a negative statement -- very likely to be true, but apparently not fulfilling your desire -- would be misplaced. | |
| Oct 21, 2020 at 21:01 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Oct 21, 2020 at 18:54 | comment | added | asrxiiviii | Thanks again. Could you please let me know what software you used for plotting and computing asymptotic behavior of this sequence? (And if possible similar alternative freely available software.) Wolfram Alpha seemed to have such a function but that doesn't work for this particular equation (namely, $n^3A_n + (n-1)^3 A_{n-2} = (34n^3-51n^2+27n-5)A_{n-1},$ $A_0:=1, A_1:= 5$) I'm working with. I apologize even though this might not seem that relevant to this forum but my only requirement for such software comes from this question, hence I feel like you can give me the best answer. | |
| Oct 20, 2020 at 17:15 | comment | added | Iosif Pinelis | @asrxiiviii : Plotting suggests that in the particular example in your (first) Edit, $A_n/C_n=O(1/n)$, just as in my counterexample. | |
| Oct 20, 2020 at 9:04 | comment | added | asrxiiviii | I see, thank you. I have provided all initial conditions in the specific example I am dealing with; I am especially interested to know at least the weaker property I added in Edit 2 for this example specified in Edits 1 and 2, since it seems to be a crucial step in a proof sketched in some lecture notes that I have been reading. | |
| Oct 20, 2020 at 2:46 | comment | added | Iosif Pinelis | @asrxiiviii : If by $A_n\sim C_n$ you mean $A_n/C_n\to1$ (as is usually done), then the above example strongly suggests that $A_n\sim C_n$ will almost never hold. If by $A_n\sim C_n$ you mean $|A_n/C_n+C_n/A_n|$ is bounded, then I think there a chance for that if $P(n)=pn^k(1+O(1/n^2))$, $Q(n)=qn^k(1+O(1/n^2))$, etc. In any case, as I said before, you need to specify appropriate initial conditions. | |
| Oct 20, 2020 at 1:22 | comment | added | asrxiiviii | By "minimal sufficient conditions" in my last question, I of course mean known results with minimal hypotheses. | |
| Oct 20, 2020 at 1:11 | comment | added | asrxiiviii | Nice! Can you also look into the above edit? Also as a curiosity, are there any minimal sufficient conditions (say on my sequence $\{A_n\}_{n=1}^\infty$ and/or polynomials appearing as coefficients etc.) guaranteeing the kind of result I want? I feel like "approximating" such linear recurrences with same degree polynomial coefficients by those with constant coefficients is something that should have been done before, but I can't find any literature. Do you know some references to that end? Thanks again! | |
| Oct 20, 2020 at 0:39 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |