What do you mean by a special function?
How to prove that gamma function is a special function ? In case of eg. expotetential integral one can reduce the problem to a linear differential equation and can ask for solvability of this equation. In this case we can use the differential algebra machinery to prove its unsolvability. But can we prove that gamma function is a special function by similar way ? As I know this function does not satisfy any simply differential equation.
The definition of the 'Gamma Function' is...
, (1)
... so that defining it as 'primitive of an elementary trascendent function' is erroneous. The [complex] derivative of (1) is...
, (2)
... which of course in not an elementary trascendent function. The reason for which the Gamma Function is 'special' is that it represents the first historical example of analytic extension of a complex function, a powerful kay that opens many 'doors' of the mathematical world ...
Kind regards