What general kinds of closed-form problems are there?

Question

A closed-form expression is a mathematical expression that contains only finite numbers of symbols and operations from a given set.
A mathematical problem is a closed-form problem if its solution is sought as a closed-form expression.

What general kinds of closed-form problems are there?

I have written below as an answer what I have found so far.

But are there other closed-form problems that are not covered by these types?

Can particular problems treated by a more general problem?

Trivially, every problem can be made to a closed-form problem - if the solution of that problem will be accepted as allowed.

Answer 1

I have written what I have found so far. Clearly some of the problems mentioned are interrelated. Some arbitrary examples of references are given here also.

a)

Usually, closed-form solutions are sought in the elementary functions, the Liouvillian functions, in terms of some special functions or in terms of Meijer G-function.

b)

• Differentiation in closed form: once, n times, infinitely often
  The function terms of the Liouvillian functions, among them the Elementary functions (Differential algebra), are infinitely differentiable. What is the state of knowledge for non-liouvillian special functions?

• Fractional Differentiation in closed form: once, n times, infinitely often
  Kleinz, M.; Osler, Th. J.: A child's garden of fractional derivatives. College Math. J. 31 (2000) (2) 82-88

• Definite integration in closed form: once, n times, infinitely often
  Bostan, A.; Chyzak, F.; Lairez, P.; Salvy, B.: Generalized Hermite Reduction, Creative Telescoping and Definite Integration of D-Finite Functions. ISSAC '18: Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation, July 2018, 95-102

• Infinite integration in closed form: once, n times, infinitely often
  That's Liouville's Integration in finite terms. Liouville's theorem and Risch algorithm can help to say which Liouvillian or elementary functions have Liouvillian or elementary antiderivatives.
  List of functions not integrable in elementary terms

• Fractional integration in closed form: once, n times, infinitely often
  Benghorbal, Mh. M.: Unified formulas for integer and fractional order symbolic derivatives and integrals of the power-inverse trigonometric class I. Int. J: Pure Appl. Math. 40 (2007) (1) 77-88

• Inverting a function in closed form
  Means calculating the inverse or partial inverses of a function in closed form.
  Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90
  Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759
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• Solving an equation in closed form
  Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50
  Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448

• Solving a difference equation in closed form
  Hendriks, P. A.; Singer, M. F.: Solving Difference Equations in Finite Terms. J. Symb. Comp. 27 (1999) (3) 239-259

• Solving a differential equation in closed form
  Singer, M. F.: Formal solutions of differential equations. J. Symb. Comp. 10 (1990) (1) 59-94

• Solving a fractional differential equation

• Solving an integral equation in closed form

• Solving an integro-differential equation in closed form

• Solving an integrodifference equation in closed form

• Solving a functional equation in closed form

• Solving a recurrence equation in closed form

• Solving a system of equations in closed form

• Solving a system of difference equations in closed form

• Solving a system of differential equations in closed form

• Solving a system of integral equations in closed form

• Solving a system of integro-differential equations in closed form

• Solving solving a system of integrodifference equations in closed form

• Solving a system of functional equations in closed form

• Solving a system of recurrence equations in closed form

• Combinations of these
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• Definite finite or infinite series in closed form
  Are there some techniques which can be used to show that a sum "does not have a closed form"?

• Indefinite finite or infinite series in closed form
  Are there some techniques which can be used to show that a sum "does not have a closed form"?

Areas where closed form solutions are of particular interest