Timeline for Is there a closed form for $\sum_{|\gamma|=k} \gamma!$
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Mar 14, 2011 at 23:55 | comment | added | Ira Gessel | A closed form is unlikely. For fixed $n$ (or $n$ small compared with $k$) the main contribution will come from the terms where for some $j$, $\gamma_j =k$ and $\gamma_i=0$ for $i\ne j$, so the sum will be asymptotic to $n\cdot k!$. More generally, asymptotic expansion can be obtained in this case by taking the terms where all but one $\gamma_i$ is a fixed integer. | |
| Mar 14, 2011 at 20:55 | comment | added | Konstantinos Panagiotou | is there any particular relation of $k$ and $n$ that interests you? | |
| Mar 14, 2011 at 15:35 | comment | added | Martin Rubey | the generating function of your sum regarded as a sequence in k seems to satisfy the relatively simple nonlinear DE $n x^2 g(x) g''(x) -(n-1)g'(x)^2 + n(3x-1)g'(x)+n^2g(x)^2=0$. Not sure whether this may be helpful, and I have no time to check right now. It may well be a triviality, in which case I beg your pardon. | |
| Mar 14, 2011 at 13:45 | answer | added | Gjergji Zaimi | timeline score: 3 | |
| Mar 14, 2011 at 12:55 | answer | added | Gerald Edgar | timeline score: 0 | |
| Mar 14, 2011 at 12:34 | history | asked | user2529 | CC BY-SA 2.5 |