Recurrence vs Recursive

Say team 1 is studying the recursive characteristics of a function.

Team 2 is studying the recurrent characteristics of the same function.

Are the 2 teams studying the same thing?

I have found for recursiveness:

"Recursive Formula

For a sequence $a1, a2, a3, . . . , an, . . .$ a recursive formula is a formula that requires the computation of all previous terms in order to find the value of an ."

http://www.mathwords.com/r/recursive_formula.htm

For recurrent-ness:

"Recurrence relation definition

A recurrence relation is an equation that defines a sequence based on a rule that gives the next term as a function of the previous term(s)."

http://mathinsight.org/recurrence_relation_definition

In case you are wondering why I am asking: I am writing an essay on the partition function. Specifically, I'll be writing on its basic recursive/recurrent? (I have seen both terms used) properties.

I would very much appreciate a explanation of any different nuances or associations the terms may hold!

Have a nice day ;)

Solution 1:

It depends upon which connotation of the terms you are referring to.

Recurrent is something that occurs often or repeatedly. However, if you are talking about a recurrence relation, then you have a mathematical structure that you are dealing with and it is certainly different than a recursive formula.

Recursion is the repeated use of a procedure or action. Generally, the procedure calls itself at some point. This differs from the definition of recurrent, in that you are strictly following a procedure or action. Recurrent can be used to define something that happens all the time, like say, rain. Recursion is also defined vaguely, whereas a recurrence relation is not. If you look up recursion on wikipedia, recurrence relations appear on the page as an example of somewhere that recursion is used.

Here is an example of a recursive function:

function factorial(n):
    if n = 1 then
        return 1
    else
        return n * factorial(n-1)
    end if
end

In fact, a recurrence relation uses recursion to define a sequence. This sequence is built in such a way that each term is defined as a combination of previous terms. The generation of such a sequence is a requirement in the definition.

Here is an example of a recurrence relation:

$$ a_1 = 1$$ $$ a_n = na_{n-1}$$

So in short. Recurrent is a word that differs from recursion. A recurrence relation uses recursion to create a sequence. Recursion is not limited to generation of sequences.