| |
 |
Generating
Functions |
| |
l |
Generating
functions are useful for manipulating sequences and therefore for
solving counting problems. |
|
Definition: |
Let S = { a0, a1, a2, a3,
... } be an (infinite) sequence of real numbers. Then the generating
function G(x), of S is the series |
| |
|
|
| |
Note: |
There is no issue of convergence here. The variable x and its powers
allow us to assign a position to the numbers ak . We will
use the closed form expressions which represent G if the series converges
but for symbol manipulation purposes only. |
| |
|
Examples:
|
| |
|
|
|
