Dealing with a double summation series

Good morning, Is there a systematic way to deal with double series? I will give to you the specific series I would like to sum symbolically but I will be also interested in some reference where I can find the technique in order to sum the following kind of expressions:

$$\sum_{m=-\infty}^\infty\sum_{n=1}^\infty z^m t^{|n|+\frac{1}{2}|n-m|+\frac{1}{2}|n+m|}$$

You can suppose that the variables are in the convergence domain of these series, I am interested in the sum or in the right way to sum these series. I can see that the powers of $t$ are symmetric when $m$ is negative, so I think that it would be interesting just the part when $m$ is positive and I think that I can deal with the other case when I have understood the procedure. If anyone (and I doubt it) is able to analytically sum these series, maybe there is a smart way to implement the calculation over a software like Mathematica. I think however that if there is an intelligent way to write the expression in order to allow a software to sum it, it is also possible to sum it by hand.

My ideas were in the direction of something like a triangular summation, but since there is both the difference than the summation of $n$ and $m$, it seems apparently difficult to define the area of summation.

Can someone help me?