pp-adic Taylor Polynomials

For a real number r≥0r≥0, we define rr-fold differentiability of a function on a pp-adic vector space by the convergence of its Taylor polynomial expansion, and compare this differentiability definition with that by iterated divided differences, the textbook approach (from the 80’s) to define pp-adic differentiability.This comparison applies to a recent definition of rr-fold differentiability over a pp-adic number field K that arises from the pp-adic Langlands program over GL2(K); yielding that this differentiability condition is equivalent to that via divided differences on K as vector space over QpQp.