Abstract： We study natural analogues of differential equations with solutions defined on functions fields over finite fields.
Analogues in this setting of important special functions such as the exponential, gamma and hypergeometric functions have been discovered by Carlitz, Goss, Thakur and others.  We will discuss integrable analogues of certain differential equations in this context and the role played by singularity analysis.  The analogues of differential equations in this context are actually discrete and yet they appear to have more in common with classical integrable differential equations rather than discrete ones. This is joint work with Jilong Zhang, Beihang University.