So I am working on a problem and I am required to convert multiple summations of the following form into a single summation,
\begin{align} \sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty} \alpha_{ijk} \,X^{i+j+k} = \sum_{n=0}^{\infty} \beta_{n} \,X^{n} \end{align}
So now I want to compare terms of the same powers of $X$ from both sides. Is there a nice way to convert the summations on LHS into a single summation? For this simple case it is easy to see that,
\begin{align} \alpha_{000}&=\beta_0\\ \alpha_{100}+\alpha_{010}+\alpha_{001} &= \beta_1 \end{align} and so on. But for some more complicated dependencies on $i,j,k$ in these summations and powers of $X$, it is not so straightforward to see this. Are there any techniques one can use? (Also for my problem differentiating both sides with $X$ is not an option.)
Any help is highly appreciated.