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).
| an=bn-1/2 | bn=an+cn-1/2 |
cn=bn/2+dn-1/2 |
| dn=cn/2+een-1/2 | een=dn-1/2 | fn=een-1/2 |
with initial conditions a0=1, b0=0, c0=0, d0=0, ee0=0, f0=0.
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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