AF:Small: Linear Differential Equations with a Convergent Integer Series Solution, Project Description

The topic in the PI’s current grant1 is: Given a linear homogeneous linear differential equation with polynomial coefficients, decide if it can be solved in terms of special functions, and if so, find such solutions. Much progress on his topic has been made by the PI and his graduate students (Sections 2 and 6). Numerous algorithms were developed that turned out to be very effective. The PI tested these algorithms on differential equations of order 2 and 3 coming from combinatorics, physics, and the OEIS (Online Encyclopedia of Integer Sequences, oeis.org). Every equation encountered that had a Convergent Integer power Series as a solution turned out to be solvable in terms of hypergeometric functions. That was a surprising observation, because the OEIS is large and contains examples from many different sources. This included dozens of equations that were not expected to be solvable in terms of hypergeometric functions. The unexpected observation leads to the following definition and question: