My take on this question, from a practical standpoint:
In the world of computers, there are no "closed forms."
"Closed form" is a mathematician's label for certain mathematical expressions which he deems "elementary." More specifically, it's a way of saying, "We don't care about the algorithms for evaluating this expression."
What makes a "closed form" is that the algorithmic steps involved in computing it are regarded in algebraic manipulation as one atomic step.
A classic example of this is the factorial, n!
. Strictly speaking, the definition involves a recurrence. However, it comes up in practice so often, mathematicians label it a "closed form" itself.
Now, perhaps some clever mathematical algorithms expert might come up with a more efficient way to compute the factorial of arbitrary values of n
, which does not involve actually performing n−1
multiplication steps. The point I'm making is that the label "closed form" doesn't depend on that better algorithm being known, or even being possible. It just means, "We are regarding the algorithm for computing this as not a crucial question (for the current text)."
In fact, if you get right down to it, f(x)=x+1
is the ultimate closed form. The unary "increment" operator.
Adding bigger numbers (the binary "sum" operator) can be seen as a "recurrence" or repeated application of the "increment" operator.
Multiplication itself is repeated application of the "sum" operator, and likewise exponentiation is a repeated application of multiplication.
But all of these are considered as "closed forms." The concept is not a fixed one.
To quote Concrete Mathematics:
We could give a rough definition like this: An expression for a quantity f(n) is in closed form if we can compute it using at most a fixed number of “well known” standard operations, independent of n. For example, 2n – 1 and n(n + 1)/2 are closed forms because they involve only addition, subtraction, multiplication, division, and exponentiation, in explicit ways.
The total number of simple closed forms is limited, and there are recurrences that don’t have simple closed forms. When such recurrences turn out to be important, because they arise repeatedly, we add new operations to our repertoire; this can greatly extend the range of problems solvable in “simple” closed form. For example, the product of the first n integers, n!, has proved to be so important that we now consider it a basic operation. The formula ‘n!’ is therefore in closed form, although its equivalent ‘1·2·. . .·n’ is not.