Discrete Functions

Recursively Defined Functions

Recursively Defined Functions

Most of the functions we have dealt with in previous chapters have been defined explicitly: by a formula in terms of the variable. We can also define functions recursively: in terms of the same function of a smaller variable. In this way, a recursive function "builds" on itself.

A recursive definition has two parts:

1. Definition of the smallest argument (usually f (0) or f (1) ).
2. Definition of f (n) , given f (n - 1) , f (n - 2) , etc.

Here is an example of a recursively defined function:


We can calculate the values of this function:

This recursively defined function is equivalent to the explicitly defined function f (n) = 2n + 5 . However, the recursive function is defined only for nonnegative integers.

Here is another example of a recursively defined function:


The values of this function are:

This recursively defined function is equivalent to the explicitly defined function f (n) = n ² . Again, the recursive function is defined only for nonnegative integers.

Here is one more example of a recursively defined function:

The values of this function are:

This is the recursive definition of the factorial function, F(n) = n! .

Not all recursively defined functions have an explicit definition.

The Fibonacci Numbers

One special recursively defined function, which has no simple explicit definition, yields the Fibonacci numbers:


The values of this function are:

Thus, the sequence of Fibonacci numbers is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … . These numbers have many interesting properties that will be studied in higher math. They recur often in mathematics and even in nature.