What does the formal definition of "closed-form" say about finite sums exactly? I have looked through the online literature and there seems to be conflicting answers to this question. Consider the finite sum
$$\sum_{i=0}^{\lfloor n/2 \rfloor} {n-1\choose i}$$
Is this expression considered closed-form?
These two links say that such a finite sum is not considered closed-form:

*

*https://opendsa-server.cs.vt.edu/ODSA/Books/CS3/html/Summations.html#:~:text=This%20is%20known%20as%20a%20closed%2Dform%20solution

*http://www3.govst.edu/wrudloff/CPSC438/CPSC438/CH05/Chapter5/Section.5.2.pdf
while this link says that such a finite sum is considered closed-form:


*https://en.wikipedia.org/wiki/Closed-form_expression#:~:text=Yes-,Finite%20sum,Yes,-Finite%20product
As far as I know, a closed-form expression has a finite number of standard operations or known functions. Based on this, my hunch is that the sum above is closed-form.
 A: Hint: The notion of closed form in mathematics is context dependent. Finite sums of type
\begin{align*}
\sum_{i=0}^{\lfloor n/2 \rfloor} {n-1\choose i}
\end{align*}
are usually not considered to be in closed form, since there is bound index variable $i$.

We find in Chapter I: What Is Enumerative Combinatorics? in the classic Enumerative Combinatorics, Vol. I by R. P. Stanley:
The basic problem of enumerative combinatorics is that of counting the number of elements of a finite set. Usually we are given an infinite collection of finite sets $S_i$ where $i$ ranges over some index set $I$ (such as the nonnegative integers $\mathbb{N}$), and we wish to count the number $f(i)$ of elements in each $S_i$ simultaneously. Immediate philosophical difficulties arise. What does it mean to count the number of elements of $S_i$? There is no definite answer to this question. Only through experience does one develop an idea of what is meant by a determination of a counting function $f(i)$. The counting function $f(i)$ can be given in several standard ways:

*

*The most satisfactory form of $f(i)$ is a complete explicit closed formula involving only well-known functions, and free from summation symbols.


Note: The crucial aspect is the term bound variable. As long as there are bound variables with limited scope given by $\Sigma$ or $\prod$ symbols, the expression is not considered to be in closed form.
