The Poincaré series of multiplier ideals of a simple complete ideal in a local ring of a smooth surface

For a simple complete ideal ℘ of a local ring at a closed point on a smooth complex algebraic surface, we introduce an algebraic object, named Poincaré series P℘, that gathers in a unified way the jumping numbers and the dimensions of the vector space quotients given by consecutive multiplier ideals attached to ℘. This paper is devoted to prove that P℘ is a rational function giving an explicit expression for it.