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 

You can add the following to your list of representations

*

*Continued fractions. See Jones & Thron or Lorentzen's books

*Integral representations (Mellin–Barnes, etc). See the ECM entry

*Exponentials.  See Knoebel's paper Exponentials reiterated

*Nested radicals. See Schuske & Thron's Infinite radicals in the complex plane

*Rational or polynomial approximation (e.g Padé approximants)

I guess the general topic is studied under Complex Analysis, Asymptotic Analysis, Harmonic Analysis.  AFAIK, there is no single book which covers all representations.
