Timeline for A classical analysis problem
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 3, 2012 at 1:10 | vote | accept | Russel | ||
| Jun 23, 2010 at 7:13 | comment | added | Wadim Zudilin | Dear Russel, from your comment above I can see that the truth for $1\le i\le 3^k$ wasn't known to you. This was my reasoning for giving more details. In order to attack your problem one needs a better formula for $x_k(t)$. I know of many unsolved problems on positivity and my reasons to do them are their serious motivation and recognised hardness. Your problem is curious, seemingly hard and that's all. Besides two of us nobody tried to help. | |
| Jun 22, 2010 at 10:40 | answer | added | Wadim Zudilin | timeline score: 3 | |
| Jun 22, 2010 at 8:55 | comment | added | Wadim Zudilin | The problem was already correct! I wonder whether you can give some motivation to study this positivity: this might be helpful. The observation I have for your sequence are: (1) $x_k(t)$ is the quotient of two polynomials of degree $2^{2k-1}$ and $2^{2k-1}-2$ for $k\ge1$ (this can be extracted from the recursion modulo irredicibility); (2) $x_k(t)-\sqrt{1-t}=O(t^{3^k})$ for $k\ge0$ and the $t$-expansion of $x_k-\sqrt{1-t}$ involves positive coefficients only. The only thing I can show is that starting with $x_0=1-t$ one gets $x_k/\sqrt{1-t}$ with negative coefficients besides the constant term | |
| Jun 22, 2010 at 7:48 | comment | added | Russel | Thanks, Zudilin, I think the statement of the problem is now clear. | |
| Jun 22, 2010 at 7:46 | history | edited | Russel | CC BY-SA 2.5 |
deleted 68 characters in body; edited tags; edited body
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| Jun 22, 2010 at 5:40 | comment | added | Wadim Zudilin | First of all, there is a "classical-analysis" tag. Secondly, renormalising $y_k(t)=x_k(t)/\sqrt{1-t}$ (so that your recurrence relation does not depend on $t$) I see that $y_k(t)$ converges to 1 pretty fast (faster than Pade approximations). This means that your sequence approached $\sqrt{1-t}$ which has all ($k>0$) coefficients negative. I just wonder whether you can give details of your construction in order to make these observations rigorous and to attack your problem. | |
| Jun 22, 2010 at 3:16 | history | asked | Russel | CC BY-SA 2.5 |