Differential equations for Fq-linear functions, II: regular singularity

We study some classes of equations with Carlitz derivatives for Fq-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 Thakurʹs 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 Fq-linear solution which vanishes on an open neighbourhood of the initial point.