# Solutions of linear ODE with INTEGER coefficeints

A meromorphic linear ODE can be solved by Frobenius method at a regular singular point. I'm interested in the solutions with integer coefficients, i.e. $$f(x)=x^\alpha\sum_{n=0}^{\infty}a_nx^n,\ \ \alpha\in\mathbb{C},\ a_n\in\mathbb{Z}.$$ I would like know what kind of linear ODE has such solutions and what kind of linear ODE does not. Does anyone know criteria?

• Whenever the recurrence relation for coefficients you get when using the Frobenius method admits for the integer solutions you have the solution with integer coefficients? – Kiryl Pesotski Jun 8 '17 at 16:00
• That's right. But the recurrence relation is not always easy. So I want other criteria, if there are any. – user356126 Jun 9 '17 at 0:22