A Simplifying a double summation

[Ad VanderVen]

TL;DR Summary

Simplifying a double summation.

Is it possible to simplify the function below so that the sums disappear.
$$\displaystyle g \left(x \right) \, = \, \sum _{j=-\infty}^{\infty} \left(-A +B \right) \sum _{k=-\infty}^{\infty} \frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(x -k \right)^{2}}{\sigma ^{2}}}~\left(U -V \right)}{\sigma ~\sqrt{\pi }}$$
with
$$\displaystyle A\, = \,1/2\,{\rm erf} \left(1/2\,{\frac { \sqrt{2} \left( -j-1/2+{\it omicron} \right) }{\rho}}\right),$$
$$\displaystyle B\, = \,1/2\,{\rm erf} \left(1/2\,{\frac { \sqrt{2} \left( -j+1/2+{\it omicron} \right) }{\rho}}\right),$$
$$\displaystyle U\, = \,1/2\,{\rm erf} \left(1/4\,{\frac { \sqrt{2} \left( -2\,bj+2\,k+1 \right) }{\tau}}\right)$$
and
$$\displaystyle V\, = \,1/2\,{\rm erf} \left(1/4\,{\frac { \sqrt{2} \left( -2\,bj+2\,k-1 \right) }{\tau}}\right)$$

 

[anuttarasammyak]

It seems very complex to me. Why do you estimate or expect that it would be reduced to a no sum form ?

 

[Ad VanderVen]

It seems very complex to me. Why do you estimate or expect that it would be reduced to a no sum form ?

I do not know.

 

[mfb]

Get rid of all the constants that can be taken out of the sums, they just blow up the expression for absolutely no reason.

You can express all the error functions as integrals and then see if adjacent terms have some nice relation for the boundaries that allows summation. The error function arguments look like there might be something you can combine.