I have a series:
$ \sum_{n_1-l_1=0}^{\infty}\sum_{n_2-l_2=0}^{\infty}\sum_{n_3-l_3=0}^{\infty}a_{n_1-l_1,n_2-l_2,n_3-l_3}r^{n_1-l_1}s^{n_2-l_2}t^{n_3-l_3} $
Which is equal to another series:
$ \sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\sum_{m=0}^{\infty}b_{k,l,m}r^{2k}s^{2l}t^{2m} $
I want to be able to equate the coefficients, but I'm a bit confused about how to change the indices in order to do so. If I set $n_1-l_1=k$ then the bases don't match ($r$ vs. $r^2$ for example). If I set $n_1-l_1=2k$ then my summation seems wrong ($\sum_{\frac{1}{2}\left(n_1-l_1\right)=0}^{\infty}$ instead of $\sum_{n_1-l_1=0}^{\infty}$).
In summary, my question is:
How can I change the indices so that I can equate the coefficients of these two equivalent series?