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.

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    $\begingroup$ Ironically, closed forms themselves are sometimes called "hypergeometric". It's a different meaning, but it might be easy to confuse. $\endgroup$ – theage Oct 15 '14 at 11:13
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    $\begingroup$ See this: math.stackexchange.com/questions/562769/… $\endgroup$ – Ron Gordon Oct 15 '14 at 11:13
  • $\begingroup$ Thanks for your comment Mr. @RonGordon, but the example in your answer there doesn't fit my OP's criterion because $$_3{\rm{F}}_2\left(\frac{1}{6},\frac{1}{2},\frac{1}{2};\frac{7}{6},\frac{3}{2};1\right)=\frac{\sqrt{\pi}\,\Gamma\left(\frac{7}{6}\right)}{\Gamma\left(\frac{5}{63}\right)}-\frac{\pi}{4}$$ $\endgroup$ – Anastasiya-Romanova 秀 Oct 15 '14 at 11:33
  • $\begingroup$ @Anastasiya-Romanova: I don't quite understand your point. The form on the left is clearly not in closed form, as it is not common knowledge how it simplifies to the closed form on the right. Again, the basic criterion I use to answer this question is, does there exist an algorithm to compute the elements of the alleged closed form faster than the integral or sum in the non-closed form? If so, then sure it is a closed form; if not, no. So the question I would have is, does there exist such an algorithm for the $_3F_2$ hypergeometric? $\endgroup$ – Ron Gordon Oct 15 '14 at 12:35
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    $\begingroup$ I suggest reading Jonathan M. Borwein and Richard E. Crandall’s nice paper Closed Forms: What They Are And Why We Care (2010). $\endgroup$ – user161303 Oct 15 '14 at 21:49

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?


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