Differential Equations for $F_q$-Linear Functions, II: Regular Singularity

Abstract

We study some classes of equations with Carlitz derivatives for $F_q$-linear functions, which are the natural function field counterparts of linear ordinary differential equations with a regular singularity. In particular, an analog of the equation for the power function, the Fuchs and Euler type equations, and Thakur's hypergeometric equation are considered. Some properties of the above equations are similar to the classical case while others are different. For example, a simple model equation shows a possibility of existence of a non-trivial continuous locally analytic $F_q$-linear solution which vanishes on an open neighbourhood of the initial point. Part I: J. Number Theory, 83 (2000), 137-154.