Are there any necessary and sufficient conditions on types of generating functions which guarantee the existence/nonexistence of a closed form for the sequence they enumerate?
Generating functions based on linear recurrence relations clearly always do (by annihilators), but are there more general statements to be made about other types of OGFs? What about DGFs and others? Does limiting the generating functions we consider to be ones obtained from different varieties of recurrence relations allow more concrete answers to be produced?
Also, if there are books, journals, or specific papers on the topic, please link them! This is fascinating stuff.