Let's say a calculus problem like an integral or a series has a solution that inevitably involving a hypergeometric function. It turns out that hypergeometric function cannot be expressed in term of certain "well-known" functions or expressions. The question then arises:
Can we consider that solution as a closed-form?
How about a solution that involving a Meijer $\rm G$-function? Please provide me an answer or a comment that contains explanations to support your arguments. I am aware that the answer of this OP can be subjective, but I would dearly love to know your thought or opinion, so please share your view about this issue as an answer or a comment. Any constructive answers or comments would be greatly appreciated. Thank you.
I believe that the study of the hypergeometric $\phantom{}_3 F_2$ function is still an active research field (suitably "modified" elliptic integrals belong to that class, for instance, and the classical elliptic integrals yet have highly non-trivial properties) while we have a good amount of knowledge about the $\phantom{}_2 F_1$ functions, so a reasonable temporary choice might be to consider the $\phantom{}_2 F_1$ functions as "elementary" and the more complex ones as "non-elementary, at the moment". However, by doing so we should name as "elementary" every Bessel/Chebyshev/Laguerre/Hermite/Legendre/Jacobi function and every spherical harmonics. On second thought, I do not think we are ready to do that: dealing with second-order differential equations introduces a whole new level of complexity, and there are many apparently easy problems that are not easy at all: for instance, how to represent a Chebyshev polynomial as a linear combination of Legendre polynomials and vice-versa?