Several sequences a0, a1, a2, ... ; b0, b1, b2, ... ; etc may be given by recurrence relations between different sequences. Here is an example (stemming from random walks) with six sequences: (note that the e-sequence is abbreviated by ee to distinguish from the number e).
with initial conditions a0=1, b0=0, c0=0, d0=0, ee0=0, f0=0.
We make use of the special character of the equations and use substitution method to get a formula for F(z): First we solve the first equation for A, , and substitute this into the second equation, solve this for B to get , substitute this into the third to get , and , and , and finally .
Take the following example:
Erich Prisner 2004