Uniform rationality of the Poincar\'e series of definable, analytic equivalence relations on local fields

Poincar\'e series of $p$-adic, definable equivalence relations have been studied in various cases since Igusa's and Denef's work related to counting solutions of polynomial equations modulo $p^n$ for prime $p$. General semi-algebraic equivalence relations on local fields have been studied uniformly in $p$ recently in \cite{16}. Here we generalize the rationality result of \cite{16} to the analytic case, unifomly in $p$, building further on the appendix of \cite{16} and on \cite{13b}, \cite{03}. In particular, the results hold for large positive characteristic local fields. We also introduce rational motivic constructible functions and their motivic integrals, as a tool to prove our main results.