Closed Form Expressions: Summation and Product Operators

Question

My question is: are expressions utilizing summation, $\Sigma$, and product, $\Pi$, operators considered 'closed-form'? To be more precise if the bounds in our summation/product operators contain variable quantities, is the expression still considered closed form?

For example, suppose that $f$ is a polynomial with real coefficients, and for every positive integer $n$, $P(n)$ is the set of all $n$-tuples $(k_1,\dots ,k_n) \in \mathbb{N_0}^n$ such that $k_1+2k_2+\dots+nk_n = n$, then is the expression $$\sum_{k \in P(n)}f(k_1+k_2+\dots+k_n)$$ considered a 'closed form expression', where $n$ is not an explicit positive integer, but say some positive integer valued variable. Say we define $g : \mathbb{N} \to \mathbb{R}$ by $$g(n) = \sum_{k \in P(n)}f(k_1+k_2+\dots+k_n)$$ and $g$ is the solution to some recurrence relation. Is $g$ a closed-form solution?

How about expressions like this? $$g(n) = \sum_{k_1=1}^{f_1(n)}\sum_{k_2=1}^{f_2(n)}\dots\sum_{k_n=1}^{f_n(n)}k_1+k_2+\dots+k_n$$ This type of expression can be rewritten as a single summation taken over a particular set whose elements depend on $n$ so it's equivalent to something like the first example, but would one only be considered closed form if the other was?

If they don't count as closed-form, are they considered analytic expressions?

I'm having trouble finding a definitive answer to these types of questions, is it really just vague and not agreed upon by the mathematical community what the precise formal definitions of things like 'closed-form expression', 'analytic expression', etc... should be?

Thank you for your time. Your input is very much appreciated.

Answer 1

The answer to your first question is: No, these are not closed forms. But as far as I know there is no standardised definition of closed-form.

It could be helpful to look what the experts tell us about it. We can read the following in section 1.2 of Concrete Mathematics written by D.E. Knuth, R.L. Graham and O. Patashnik:

D. Knuth, et al.: ... Incidentally we've been talking about closed forms without explicitely saying what we mean. Usually it's pretty clear.

Recurrences like \begin{align*} T_0&=0\\ T_n&=2T_{n-1}+1\qquad\qquad n>0 \end{align*} are not in closed form - they express a quantity in terms of itself; but solutions like \begin{align*} T_n=2^n-1\qquad\qquad n\geq 0 \end{align*} are. Sums like $$1+2+\cdots+n$$ are not in closed form - they cheat by using $\ldots$; but expressions like \begin{align*} \frac{n(n+1)}{2} \end{align*} are.

We could give a rough definition like this: An expression for 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, $2^n-1$ 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 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 $1\cdot 2\cdots n$ is not.