Semantic meaning of 'finiteness' in closed forms

Question

Definition by Wolfram MathWorld: "An equation is said to be a Closed-form Solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set. For example, an infinite sum would generally not be considered closed-form. However, the choice of what to call closed-form and what not is rather arbitrary since a new "closed-form" function could simply be defined in terms of the infinite sum."

Question: Should we interpret finiteness of sum considered as closed form by terms of nessesarity or primality ? To articulate what I mean, let us consider this example.

Example: Let us assume that we want closed form of function $f(x)=(x+420)^{69} $ as sum of power functions in terms of natural powers with some coefficients. By using of binomial theorem , we know that:

$ \displaystyle (x+420)^{69}=\sum_{n=0}^{69} \binom {69}{n}x^n 420^{69-n} $

In the case of nessesarity , the series above would be considered as closed form because its finiteness is minimum condition of existence (so it have to be symbolically reduced to its nessesarity if that is possible). In that semantic interpretation , sum from $0$ to some natural integer $k $ greater than 69 would not be considered as closed form, even if that representation is true and finite.

In the case of primality , the series above would be not considered as closed form because its finiteness is secondary in respect of the infiniteness (so it have to be symbolically expand to its primality if that is possible). In that semantic interpretation , sum from $0$ to $69$ would be consider as closed form only if for any integer $k $ greater than 69 series would give representation that is untrue.

Note from the OP/Author: Question was asked as initiator of discussion and as highlighter of problem. There is no such thing as incorrect answer (until there is some official/formal statement about that) OP/Author personally thinks after Emanuel Kant that Mathematics is apriori , so we should use condition of primality , but in case it is aposteriori we could use condition of nessesarity but OP/Author guesses that there is no such problem.

Another example (here are full calulations https://math.stackexchange.com/a/4622585/1117198)

$\displaystyle \int_0^\infty \frac{t^s}{(e^t-1)^z}dt = \frac {sin (\pi z)\Gamma (s+1)}{\pi}\left [\zeta (s+2,z) -\zeta (s+1,z)(\psi^0( 1 - z) + \gamma)... \right]$

Series above is infinite series but for natural $z$ everything reduce to finite sum. And I'm arguing about, is it a closed form. And I think that this finite form for natural $z$ is indeed the same 'class' as this infinite one. To be precize, I think that this finite form is truly infinite series but it just reduces, so if you accept that as closed form you have to acept infinity series as closed form too. So I write this post becouse it is interesting topic and I am greedy so I want to have accepted anwser XD

Answer 1

I think OP is asking about 2 things. I will try to give my thoughts on that.

(P) What is Closed-form Solution ? (Q) Does it necessarily involve finite terms ?

(1) Summation "$x=0.1+0.01+0.001+0.001 + \cdots$" is infinite , though the Equivalent "$x=1/9$" is finite.
The infiniteness is not necessary here.

(2) Summation "$\pi=3+0.1+0.04+0.001+0.0005+0.00009 + \cdots$" is infinite. The infiniteness is necessary here , we have no way to write it with finite Decimal terms.

(3) With trigonometry , "$\pi=2\sin^{-1}(1)$" is finite. It is arbitrary to say that this is Closed form.

(4) Given Constants $u$ & $v$ , Equation "$x=\sin(ux)-\cos(vx)$" may have a Solution , though we do not have a Closed-form Solution.
We might get some Solution "$x=\Pi[\Sigma\text{complicated terms in Powers of u & v}]$" which might still not be Closed-form Solution.
We might make some new function to get "$x=sicouv(u,v)$" , which we might arbitrarily call a Closed-form Solution.
This is what Wolfram (& user lulu) state , about making a new function to hide the Details & the infiniteness.

(5) We have no Direct way to count the number of relatively Prime numbers , hence $\phi(n)$ the "totient function" is not in closed form , even though no infinite series is involved.
With that, we have $\Phi(n)$ the "Totient Summatory Function" looking like Closed form , which is quite arbitrary , where the individual terms are not known.

(6) Consider "$P(x)=\sum_{y=0}^{n}{[[f(y)]]}$" , where $n$ is finite.
We can arbitrary call this Closed form (when our readers/audience agree) , though that is not a Precise term.
Let us say $f(y) \equiv 0$ when $y$ is larger than $n$ : Then we get "$P(x)=\sum_{y=0}^{\infty}{[[f(y)]]}$" : This infinity is not necessary. Still , we might arbitrarily say that this is Closed form , while the finite way exists.

In short , to Answer the 2 Questions :
(P) What is Closed-form Solution ? (Q) Does it necessarily involve finite terms ?
(P) : It is arbitrary to call something Closed form , it is some agreement between writer/author & audience/reader. When something is in terms of Popular well-known functions , it might be Closed form.
(Q) The finiteness/infiniteness is not a general concern/criteria in the classification.
There are Closed form functions with necessarily infinite terms.
There are Closed form functions where infinite terms are not necessary & can be eliminated.
There are Closed form functions with finite terms.
Some functions with necessarily infinite terms may not be Closed form.
Some functions with unnecessary/eliminatable infinite terms may not be Closed form.
Some functions with finite terms may not be Closed form.
Classification is with Popularity & well-knownness , not with finiteness & infiniteness.
The Classification will not change the Properties / Evaluations / Intricacies of the functions.

Answer 2

Let's define that $f(x)$ has closed form solution, only if its representation is finite sum of axiomated set of well-known functions $<g_0(\xi),...,g_i(\xi)>$ in the form $f (x)=\sum_{n=0}^k h_i (x,n) \wedge k \in \mathbb{N}$, where $h_i (x,n)$ is made of natural amount of convolutions of set of well-known functions. Let's define some order of set $<g_0(\xi),...,g_i(\xi)>$, that $\forall_{j <i} g_j(\xi)= \sum_{n=0}^{k} h_{j-1}(\xi,n) \wedge \left (k \notin \mathbb{N} \vee k \rightarrow \infty \right)$ where $h_{j-1}(x,n)$ is naturally convoluted from the set $<g_0(\xi),...,g_{j-1}(\xi)>$

We would call the sum to be finite in terms of necessity, only if its series representation will be symbolically reduce to its minimum conditions of existence. $\sum_{n=0}^k h (x,n)=f (x) \wedge \sim \exists_{k_0 \in \mathbb{N} <k } \sum_{n=0}^{k_0} h (x,n)=f (x) $. In such definition function $f (x)$ would have some infinite series representation only if $\sum_{n=0}^{\infty} h (x,n)=f (x) \wedge \sim \exists_{k_0 \in \mathbb{N} } \sum_{n=0}^{k_0} h (x,n)=f (x) $

We would call the sum to be finite in terms of primality, only if its series representation will be symbolically expanded to its most general condition of existence. $\sum_{n=0}^k h (x,n)=f (x) \wedge \forall_{k_0 \in \mathbb{N} >k } \sum_{n=0}^{k_0} h (x,n) \neq f(x) $. In such definition function $f (x)$ would have some infinite series representation $\sum_{n=0}^{\infty} h (x,n)=f (x) $

We call some finite or infinite series as strong, if conjunction of its necessity and primality is true, so its representation is uniquely defined. It add some new function to set of well-known functions by making $k $ no natural. Proof is trivial from definition. If it would have closed form it would have some finite sum representation. Strong infinite series from definition is not closed form too. In such way $f (x)$ can be written as $g_{i+1}(\xi)$ and added to extant set of well-known functions.

It can't be say about weak series which is finite by necessity and infinite by primality. It will be just some special case of more general function naturally convoluted from set of well-known functions. Prove is trivial too. Coefficient of all type are naturally convoluted from set of well-known functions so they are dependent from $n $. Series is truly infinite but coefficients above $k $ vanishes. If finiteness is defined by necessity and infiniteness by primality, from one side series is narrowed as closed form and from another as some new function which from definition is not yet in set of well know-functions. This is contradiction so only one can be true but we just prove that this is closed form by writing it as finite sum from set of well-known, so for some no natural $k $ we get just some $f (x,k)$. If some function $g_{j}(\xi)$, $j <i $ could be written as weak series from set of well-known functions $<g_0(\xi),...,g_{j-1 }(\xi)>$ it can be reduced from set $<g_0(\xi),...,g_{i }(\xi)>$