Different ways to represent functions other than Laurent and Fourier series? In the book "A Course of modern analysis", examples of expanding functions in terms of inverse factorials was given, I am not sure in today's math what subject would that come under but besides the followings : power series ( Taylor Series, Laurent Series ), expansions in terms of theta functions, expanding a function in terms of another function (powers of, inverse factorial etc.), Fourier series, infinite Products (Complex analysis) and partial fractions (Weisenstein series), what other ways of representing functions have been studied? is there a comprehensive list of representation of functions  and the motivation behind each method?
For example , power series are relatively easy to work with and establish the domain of convergence e.g. for $ \sin , e^x \text {etc.}$ but infinite product representation makes it trivial to see all the zeroes of $\sin, \cos \text etc. $ 
Also if anyone can point out the subject that they are studied under would be great.
Thank you 

There are literally dozens of ways to represent "arbitrary" (given or unknown) functions $f$ in terms of "special" functions. Each of these ways responds to the particular geometrical situation at hand, to ways of encoding the available information about $f$, to a-priori-conditions that $f$ must fulfill, etc. The special functions used in each particular case are "taken from a catalogue", they are well understood and usually have an algebraically describable behavior with respect to the natural operations (differentiation, shifts, rotations, etc.) present in the given environment.
