Subject: Recursive definitions and Derivatives
From: "William J. Rapaport" <rapaport@buffalo.edu>
Date: Tue, 17 Nov 2009 11:25:41 -0500 (EST)

A student writes:
"I can't help noticing a similarity between the recursive definition of
a discrete function, and the derivative of a continuous function.
Both describe the change in the function (the recursive definition and
the derivative both tell you the change between one output and its
successor), and they both require an initial condition in order to find
the original equation (when integrating, you need the initial condition
to solve for the constant). In fact, the example you went over in class
followed the rules of a derivative: the function was linear, and its
recursive definition was a constant. Is the recursive definition of a
discrete function analogous to the derivative of a continuous function?
A better question might just be what is the difference between them?

Also, can y=sin(x) be represented by a discrete function?
If so, what is its recursive definition?"

Reply:
Yes; they are not only similar, but they are intimately related.
The short reason is that, just as derivatives are differences between
values of a continuous function, the way we are computing (some)
recursively defined functions, they are what are called "finite"
differences. (That's why the 5 in 5n+1 is related to the 5 in h(n)+5.
Think about what the derivative of 5n+1 is.) For more on this topic,
take a look at some parts of the Rosen text. Unfortunately, as I write
this, I don't have the text at hand. I'll give you page references
tonight or tomorrow.

As for your question about sin(x), I really don't know the answer
(I'm a discrete mathematician, not a continous one :-) but my intuitions
tell me that there would be some kind of discrete analog of the sin
function, and that sin should be recursively definable. I'll look into
it to see what I can find out.

Good questions!