Timeline for Asymptotics of the unique root of a polynomial equation defined as a sum of rational expressions
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Feb 16, 2023 at 20:02 | comment | added | dohmatob | Ok thanks for clarifying, and glad you find the question vaguely interesting. | |
| Feb 16, 2023 at 13:39 | comment | added | Claude Leibovici | @dohmatob. The polygamma functions and generalized harmonic numbers. I shall have a look at $\lambda_i=i^{-\beta}$. If anythig interesting, I shall edit. By the way, I faced very similar problems ... 50+ years ago ! You made me feeling youger. Cheers :-) | |
| Feb 16, 2023 at 13:34 | comment | added | dohmatob | Thanks for the input. That's an interesting approach. What are the $\psi$'s and $H$'s ? BTW, along the lines of your approach: if $\lambda_i \asymp i^{-\beta}$ for some fixed $\beta \in (1,\infty)$, then one can (looking at integrals instead of sums) show that $f(t) \asymp t^{1-1/\beta}$. In this case, we then have the analytic formula $t_\star \asymp \epsilon^{\beta/(\beta-1)}$, where $\epsilon := \sqrt{s/M}$. | |
| Feb 16, 2023 at 12:51 | history | answered | Claude Leibovici | CC BY-SA 4.0 |