Does this sum have a closed form?

Question

empty · Answer

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

amistre64 · Answer

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

empty · Answer

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

amistre64 · Answer

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

amistre64 · Answer

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

amistre64 · Answer

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

amistre64 · Answer

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

amistre64 · Answer

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

empty · Answer

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.

amistre64 · Answer

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

amistre64 · Answer

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}]

amistre64 · Answer

firefox doesnt like to paste the links correctly ..

amistre64 · Answer

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

amistre64 · Answer

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

amistre64 · Answer

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

empty · Answer

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

thomas5267 · Answer

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

amistre64 · Answer

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