Bernstein–Sato polynomials in positive characteristic

In characteristic zero, the Bernstein–Sato polynomial of a hypersurface can be described as the minimal polynomial of the action of an Euler operator on a suitable D-module. We consider analogous D-modules in positive characteristic, and use them to define a sequence of Bernstein–Sato polynomials (corresponding to the fact that we need to consider also divided powers Euler operators). We show that the information contained in these polynomials is equivalent to that given by the F-jumping exponents of the hypersurface, in the sense of Hara and Yoshida [N. Hara, K.-i. Yoshida, A generalization of tight closure and multiplier ideals, Trans. Amer. Math. Soc. 355 (2003) 3143–3174].