Here we are concerned with the question of whether or not a given problem, such as the generating function for a combinatorial problem is solvable. What does it mean for a problem to be solvable? This depends on ones point of view. We may seek a solution in terms of known functions (algebraic, hyper-geometric etc.). If this is not achievable we say the problem is `solved' if we can express the function as a solution of a differential equation or a functional equation from which we can extract exact information about the problem (location of singularities, values of critical exponents and so on).
Of course we hope that ultimately all problems are solvable, we just have to wait for suitable mathematical methods to be developed.
Enting-Guttmann Solvability Test
Many classical problems in statistical mechanics and combinatorics have solutions in a class of functions called D-finite. Briefly this means that the function is a solution to a differential equation of finite order with polynomial coefficients.
Enting and Guttmann have developed a numerical test which can help us quickly determine whether or not a given function may be D-finite.