Closed Form Solutions for Linear Differential and Difference Equations

Finding closed form solutions of differential equations has a long history in computer algebra. For example, the Risch algorithm (1969) decides if the equation y′ = f can be solved in terms of elementary functions. These are functions that can be written in terms of exp and log, where “in terms of” allows for field operations, composition, and algebraic extensions. More generally, functions are in closed form if they are written in terms of commonly used functions. This includes not only exp and log, but other common functions as well, such as Bessel functions or the Gauss hypergeometric function. Given a differential equation L, to find solutions written in terms of such functions, one seeks a sequence of transformations that sends the Bessel equation, or the Gauss hypergeometric equation, to L. Although random equations are unlikely to have closed form solutions, they are remarkably common in applications. For example, if y = ∑︀∞ n=0 anx n has a positive radius of convergence, integer coefficients an ∈ Z, and satisfies a second order homogeneous linear differential equation L with polynomial coefficients, then L is conjectured to be solvable in closed form. Such equations are common, not only in combinatorics, but in physics as well. The talk will describe recent progress in finding closed form solutions of differential and difference equations, as well as open questions.