Analytic Solution of Poiseuille Flow

Hey guys,
I happened across the paper “Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model” by He et al., and followed the derivation up until equation 9. But can’t seem to figure out how to make the jump from equation 8 to 9, I’m sure it’s probably really simple when you know how. Does anyone have an idea how its done?
Thanks in advance,

Please tell us if you do not understand the first or the second equal sign in Eq. (9).

Hey Timm,
I can see where the first part of the equation is from. But when I substitute in Eq. (8) I cant seem to get a closed form for Eq. (9), because the distribution functions from 5 to 8 all have a term involving the same distribution function on the adjacent node.
Any ideas?

I see the problem. On first gaze, I do not see how it is done either, but I am pretty sure that somehow the recursiveness is used. This also explains why the solution Eq. (9) is only valid for 1 < j < n - 1. Try to plug in, e.g., f^{j+1}_7 into f^j_7 and so on and see what happens.