Timeline for Log-convexity of Lassalle's sequence
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 3, 2021 at 22:32 | comment | added | T. Amdeberhan | @JoeSilverman and Pietro: thank you both for interesting comments. | |
| Mar 3, 2021 at 21:59 | comment | added | Pietro Majer | According to a comment on the OEIS link (by Sergei N. Gladkovskii) $$\sum_{k\ge0} A_k\frac{x^{2k+2}}{(2k+2)!} = -\log u(x),$$ for $ u(x):= \frac{J_1(2x)}x$. This function satisfies an even simpler linear ODE than the Bessel function, namely $x\ddot u+3\dot u+4xu=0$. (But composing with $\log$ produces a nonlinear quadratic ODE, as suggested by the Cauchy product in the inductive definition). | |
| Mar 3, 2021 at 21:30 | comment | added | Joe Silverman | More experiments: Let $$B_n=\frac{A_n+1}{A_n}-\frac{A_n}{A_{n-1}}$$, so you want to prove that $B_n\ge0$. Experimentally, it looks as if $$\lim_{n\to\infty} B_{n+1}-B_n = \beta\quad\text{for some $\beta\approx2.1795$.} $$ | |
| Mar 3, 2021 at 21:24 | comment | added | Joe Silverman | I assume you checked the first few $n$. For the small list in OEIS, it looks as if $\log(A_{n+1}A_{n-1}-A_n^2)$ is growing fairly rapidly, very roughly like $\log(n!)$. | |
| Mar 3, 2021 at 20:08 | history | asked | T. Amdeberhan | CC BY-SA 4.0 |