Generating Functions: The Basics
A formal power series
is an expression of the form
A generating function is a power series formed from a sequence
Important Power Series Identity (ID1):
Methodology:
1) Define generating function
from sequence
2) Multiply both sides of recurrence by
and sum over all
3) Express both sides of identity in terms of
and solve for
4) Expand
as a power series
Example 1: Fibonacci Sequence
Finding the general term:
Define
. Then:
.
Define also
Then:
Therefore, we have that
Here is a more practical application:
Example 2: What is the generating function for a partition of
integers?
Solution:
into a sum of even positive