Summation and integral representations of special functions I have a slightly strange (and possibly quite vague) question that I'm keen to hear people's thoughts on. In my recent work, I have been coming across various infinite series and integrals that cannot be evaluated analytically (at least Mathematica doesn't evaluate them). For example, see Summation of a quotient with a square root.
Ideally, I'm thinking that there should be special function representations for these, but maybe they just haven't been invented yet? Is it reasonable to seek out possible routes for getting special functions defined by such representations such that their properties can be found (by suitable experts i.e. probably not me!) and how would I go about doing this? Is this how some special functions have been defined/invented historically? If Mathematica (or an equivalent) could then have these as built in functions, it would make my notebooks run far quicker. Am I being crazy thinking this?
 A: My answer below is far to be a definitive answer. Better, it can be consideredas as a  too long comment.
You are absolutely right to say that  << Ideally, I'm thinking that there should be special function representations for these, but maybe they just haven't been invented yet? >>. Nevertheless "inventend" is questionable to mean "studied, described, published, referenced, standardized,... ". 
I would like to cite : << A special function has to acquire a background of property, descriptions, formulas and derivations as extended as possible. Before becoming a referenced special function, its name has to be spread in the literature in order to become familiar. More importantly, the function should be useful in a branch of mathematics or physics >> ( From page 3 in https://fr.scribd.com/doc/34977341/Sophomore-s-Dream-Function ). 
A discussion about the use of special functions  : pp.18-36 in https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales . (Paper for general public).
A list of special functions : https://en.wikipedia.org/wiki/List_of_mathematical_functions#Other_standard_special_functions
Your question : << Is it reasonable to seek out possible routes for getting special functions defined by such representations such that their properties can be found... ? >>.
This is a big question which is far to be fully solved. In fact this question cannot be raised without to specify the list of special functions to be considered in the search. 
From precursor a long time ago by Joseph Liouville and others : https://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra)
A relevant answer is : https://en.wikipedia.org/wiki/Risch_algorithm
The method used in WolframAlpha and other softwares of this kind are variants of the Risch_algorithm. 
