Closed Form Solution for Series with Exponential and Power Terms?

[rman144]

\sum_{n=1}^{\infty}(-1)^{n}\frac{e^{-(nx)^2}}{n^{1-m}}

Where m is an integer and 0<x<oo. I need a closed form solution, and was thinking something along the lines of a theta-type function, but cannot seem to locate any identities that match. Anyone have a suggestion?

 

[jasonRF]

rman144,

I'm curious what you consider "closed form" and what you will use it for (analysis, numerics, etc.). Clearly, for x>1 this is an excellent representation that converges quickly - I would be surprised if a theta function (or hypergeometric or a G-function or ...) would be any "simpler" in reality or easier to compute, although on paper you may be able to write a small number of special functions, at best. Of course for x<1 the series may leave something to be desired, as I am pretty sure it can have a large number of terms with increasing magnitude before the terms start to decrease.

Just curious.

Jason

 

[jasonRF]

rman144,

I just re-read my post - it sounds like I am questioning the utility of finding another representation of the series. I didn't mean it that way - honest! I'm guessing most of us have been in a similar situation of looking for a nicer representation that may yield more insight, allow us to use well documented properties of known functions, allow us to use code we already have to compute it, etc. Anyway, I really am curious about the source of the series. Also, are there any other constraints on m beyond being an integer (even/odd, positive/negative)?

Regards,

Jason

 

[Gib Z]

Perhaps replace the exponential by its Taylor series, switch the double sums? It's something to try.

 

[rman144]

Lol, I took no offense. Thank you for the help.