Several sequences a_{0}, a_{1}, a_{2}, ... ; b_{0},
b_{1}, b_{2}, ... ; 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).

a_{n}=b_{n-1}/2 |
b_{n}=a_{n}+c_{n-1}/2 |
c |

d_{n}=c_{n}/2+ee_{n-1}/2 |
ee_{n}=d_{n-1}/2 |
f_{n}=ee_{n-1}/2 |

with initial conditions a_{0}=1, b_{0}=0, c_{0}=0,
d_{0}=0, ee_{0}=0, f_{0}=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