## Empty one year ago Does this sum have a closed form?

1. Empty

$\sum_{n=0}^\infty (2n+1) e^{-k(n^2+n)}$ where k > 0 is a real number.

2. amistre64

when i take the integral, i get something close to the closed form.

3. Empty

Hmmm, how could I go about finding the error of the approximation? I am fine with this idea this sounds pretty good.

4. amistre64

$c_n~e^{-k(n^2+n)}$ gives me the 1/e stuff, but my coefficients are not proper ...

5. amistre64

i wouldnt know off hand how to find the error stuff. its been a long day

6. amistre64

k=1; 3,10,21,36,55,78,... k=2; 3,10,21,36,55,78,... k=1/2; same progresion

7. amistre64

3,10,21,36,55,78 7 11 15 19 23 4 4 4 4 3+7n+4n(n-1)/2 seems to cover that ... but thats starting at n=0 so a shift is needed

8. amistre64

heh, 2n^2+n is the simplified shift ..

9. Empty

Yeah that's totally fine, I was thinking more or less of solving them independently to within a constant or something I'm not too concerned this is for my own interest in looking for the closed form of the molecular partition function of a linear rigid rotor. I noticed it doesn't have a closed form and the book I'm reading says to evaluate it numerically but I thought that might be kind of lame and wanted a closed form haha.

10. amistre64

with a little help from the wolf tech ... $\sum_{n=0}^{\infty} (2n+1)e^{-k(n^2+n)} =(2n^2+n) e^{-k(n^2+n)}$ it works for a couple of test values for k

11. amistre64

http://www.wolframalpha.com/input/?i=table [%282n^2%2Bn%29e^%28-2%28n^2%2Bn%29%29%2C{n%2C10}] http://www.wolframalpha.com/input/?i=table [sum%28i%3D1+to+n%29+%282n%2B1%29e^%28-%28n^2%2Bn%29%2F2%29%2C{n%2C10}]

12. amistre64

firefox doesnt like to paste the links correctly ..

13. amistre64

i notice the shift in the polynomial part, like a usual difference equation. $(1+2n) +0n^2+0n^3+...\color{red}{\implies} 0+(1n+2n^2)+0n^3+...$

14. amistre64

not sure how useful my prattling is, but it looked fun to try out is all :)

15. amistre64

the setup doesnt work for n=0 tho ...

16. Empty

That's fine I think this helped actually haha thanks xD

17. thomas5267

Mathematica returned nothing useful. Looks like it doesn't have a close form.

18. amistre64

i noticed an error in my plans ... my table indexed i, from 0 to n .... but i forgot to use i in the process :/ oh well. it still looks fun tho