Timeline for Does this expression always vanish?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 24, 2023 at 23:10 | comment | added | Michael Engelhardt | @quarta - no, it's not that simple. The product is not symmetric under exchange of $i$ and $j$. | |
| Sep 24, 2023 at 20:08 | comment | converted from answer | quarta | This is a comment, not an answer but I am not entitled. By exchanging $i$ and $j$ one sees that each term in the double sum occurs twice, with alternating sign. | |
| Sep 24, 2023 at 3:14 | vote | accept | Silly Point | ||
| Sep 24, 2023 at 3:13 | vote | accept | Silly Point | ||
| Sep 24, 2023 at 3:14 | |||||
| Sep 23, 2023 at 12:29 | history | became hot network question | |||
| Sep 23, 2023 at 9:24 | answer | added | Ilya Bogdanov | timeline score: 35 | |
| Sep 23, 2023 at 8:46 | comment | added | Per Alexandersson | There should be some symmetric function identity hiding here. | |
| Sep 23, 2023 at 6:59 | comment | added | T. Amdeberhan | It seems equivalent to \begin{align} \sum_{i=1}^N\sum_{j=1\\ j\ne i}^N\frac{A_i+A_j}{A_i-A_j}\prod_{k=1\\ k\ne i}^N\frac{A_i A_k}{(A_i-A_k)^2}=0. \end{align} | |
| Sep 23, 2023 at 6:46 | comment | added | Greg Martin | Other observations: the expression can be written as $$(A_1\cdots A_n) \sum_{1\le i<j\le n} \frac{A_i+A_j}{(A_i-A_j)^3} \biggl( A_i^{n-2} \prod_{\substack{1\le k\le N\\k\ne i\\k\ne j}}\frac1{(A_i-A_k)^2} - A_j^{n-2} \prod_{\substack{1\le k\le N\\k\ne i\\k\ne j}}\frac1{(A_j-A_k)^2} \biggr).$$This makes it easy to see that the order of the pole at any $A_i=A_j$ is at most $2$ (multiplying through by $(A_i-A_j)^3$ and setting $A_i=A_j$ yields $0$). | |
| Sep 23, 2023 at 6:41 | comment | added | Carlo Beenakker | as a simple test, if I set $A_i=i$ for all $i$, the expression can be evaluated in closed form and gives 0. | |
| Sep 23, 2023 at 4:54 | comment | added | David Roberts♦ | On MathOverflow, question titles that are commands (like "Prove that...") give the impression that the original source is a homework or textbook question. It's best to phrase the title as an actual question, not something that gives the impression you know the answer and you're challenging users here. | |
| Sep 23, 2023 at 4:52 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
removed imperative tense from title
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| S Sep 23, 2023 at 4:26 | review | First questions | |||
| Sep 23, 2023 at 5:57 | |||||
| S Sep 23, 2023 at 4:26 | history | asked | Silly Point | CC BY-SA 4.0 |