I'm currently reading "The cumulant lattice Boltzmann equation in three dimensions: Theory and validation" from Geier et. al. and have some trouble in a proof.
We have given multivariat cumulants (normalized with $m_{00}$) in terms of moments, like the following:
$$ K_{11} = m_{11} - \frac{m_{10}m_{01}}{m_{00}} $$
We can expand $K$ for expansion parameter $\epsilon$ to get our series expansion \begin{equation} K_{11} = \sum_{i=0}^\infty \epsilon^i K_{11}^{(i)} \end{equation}
If I now want to compute, let's say, $K_{11}^{(2)}$ in terms of expanded moments, I'd say we have to do
\begin{equation} \begin{aligned} K_{11} & = m_{11} - \frac{m_{10}m_{01}}{m_{00}}\\ \Leftrightarrow \sum_{i=0}^\infty \epsilon^i K_{11}^{(i)} & = \sum_{i=0}^\infty \epsilon^i m_{11}^{(i)} - \frac{\sum_{i,j=0}^\infty \epsilon^{i+j} m_{10}^{(i)}m_{01}^{(j)}} {\sum_{i=0}^\infty \epsilon^i m_{00}^{(i)}} \end{aligned} \end{equation} and match the $\epsilon$ coefficient on both sides.
In the paper, he states \begin{equation} K_{11}^{(2)} = m_{11}^{(2)} - \frac{m_{10}^{(1)}m_{01}^{(1)}} {m_{00}^{(0)}} \end{equation} Which obviously has same $\epsilon$ coefficients on both sides, but I can't see, why this is the right way to do it?
Edit: thanks for pointing to the geometric series.
The sum in the denominater can be dealt with a geometric series, like follows \begin{equation} \begin{aligned} K_{11} & = m_{11} - \frac{m_{10}m_{01}}{m_{00}}\\ \Leftrightarrow \sum_{i=0}^\infty \epsilon^i K_{11}^{(i)} & = \sum_{i=0}^\infty \epsilon^i m_{11}^{(i)} - \frac{\sum_{i,j=0}^\infty \epsilon^{i+j} m_{10}^{(i)}m_{01}^{(j)}} {\sum_{i=0}^\infty \epsilon^i m_{00}^{(i)}}\\ \Leftrightarrow \sum_{i=0}^\infty \epsilon^i K_{11}^{(i)} & = \sum_{i=0}^\infty \epsilon^i m_{11}^{(i)} - \frac{\sum_{i,j=0}^\infty \epsilon^{i+j} m_{10}^{(i)}m_{01}^{(j)}} {m_{00}^{(0)}} \frac{1}{1 - \sum_{i=1}^\infty \epsilon^i \frac{ - m_{00}^{(i)}}{ m_{00}^{(0)}}}\\ \Leftrightarrow \sum_{i=0}^\infty \epsilon^i K_{11}^{(i)} & = \sum_{i=0}^\infty \epsilon^i m_{11}^{(i)} - \frac{\sum_{i,j=0}^\infty \epsilon^{i+j} m_{10}^{(i)}m_{01}^{(j)}} {m_{00}^{(0)}} \sum_{j=0}^\infty {\left(\sum_{i=1}^\infty \epsilon^i \frac{ - m_{00}^{(i)}}{ m_{00}^{(0)}}\right)}^j \end{aligned} \end{equation} Which leaves the second order terms \begin{equation} \begin{aligned} K_{11}^{(2)} &= m_{11}^{(2)} \\ &\quad- \frac{ m_{10}^{(1)}m_{01}^{(1)} + m_{10}^{(0)}m_{01}^{(2)} + m_{10}^{(2)}m_{01}^{(0)} }{m_{00}^{(0)}}\underbrace{1}_{j=0}\\ &\quad +(m_{10}^{(1)}m_{01}^{(0)} + m_{10}^{(0)}m_{01}^{(1)})\underbrace{ \frac{m_{00}^{(1)}}{m_{00}^{(0)2}}}_{j=1,i=1}\\ &\quad + m_{10}^{(0)}m_{01}^{(0)} \left( \underbrace{\frac{m_{00}^{(2)}}{{m_{00}^{(0)2}}}}_{j=1,i=2} - \underbrace{\frac{m_{00}^{(1)2}}{m_{00}^{(0)2}}}_{j=2,i=1} \right)\\ &= m_{11}^{(2)} \\ &\quad- \frac{ m_{10}^{(1)}m_{01}^{(1)} + m_{10}^{(0)}m_{01}^{(2)} + m_{10}^{(2)}m_{01}^{(0)} + m_{10}^{(0)}m_{01}^{(0)}m_{00}^{(1)2} }{m_{00}^{(0)}}\\ &\quad + \frac{m_{00}^{(1)}(m_{10}^{(1)}m_{01}^{(0)} + m_{10}^{(0)}m_{01}^{(1)} - m_{10}^{(0)}m_{01}^{(0)}m_{00}^{(1)})}{m_{00}^{(0)2}} \\ &\quad + \frac{m_{00}^{(2)}m_{10}^{(0)}m_{01}^{(0)}}{m_{00}^{(0)2}} \end{aligned} \end{equation} Why should I not be allowed to match $*^{(0)}$ and $*^{(2)}$. Sure, they are on different scales, but I suppose they can be combined, anyway. Do I miss something, or is there additional modelling work to be done to get rid of those terms? For reference, this is equation (G.29) and following in the paper. I could just accept to not mix them, but I would like to understand whats happening here.
Thanks
ps: I am not allowed to create new tags, and the ones I found do not match the problem, sorry about that
I mailed the author about it, and got a super fast response:
The key is, that I can choose my frame of reference once in the analysis, and to simplify the calculations, one chooses it in this case such that $m_{10}=0=m_{01}$.
Much more easy than I thought.