ABSTRACT
This chapter reviews a taxonomy of series and analytic functions and considers the induced classification on combinatorial classes. It describes some useful analytic properties, closure properties and characterizations of the combinatorial classes whose generating functions lie in that class. The chapter covers the following functions: algebraic functions, D-finite functions, closure properties and G-functions. It focuses on Combinatorial classes with D-finite generating functions and combinatorial classes with non-D-finite generating functions. A D-finite function can be naturally encoded by a differential equation or system of differential equations with uniqueness assured by a finite set of initial conditions. Algebraic and transcendental functions are a very classic topic of mathematics. Diagonals of rational functions are the largest source of transcendental D-finite functions in combinatorics.
