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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?

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The second series only has terms where $r$, $s$, and $t$ are all raised to even powers. So the terms in the first series where any of them is raised to an odd power must vanish. That is, you must have $a_{x,y,z}=0$ unless $x,y,z$ are all even; and in that case $a_{x,y,z}=b_{x/2,y/2,z/2}$. – mjqxxxx Jan 17 '12 at 16:33
up vote 2 down vote accepted

Since the first series is $$\sum_{k=0}^{\infty}\sum_{\ell=0}^{\infty}\sum_{m=0}^{\infty}a_{k,\ell,m}r^{k}s^{\ell}t^{m}, $$ the two series coincide if and only if $a_{2k,2\ell,2m}=b_{k,\ell,m}$ for every nonnegative $k$, $\ell$ and $m$, and $a_{k,\ell,m}=0$ as soon as $k$, $\ell$ and $m$ are not all even.

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