Our purpose in this monograph is to provide a concise and complete introduction to the study of arithmetic differential operators over the p-adic integers ℤp. These are the analogues of the usual differential operators over say, the ring ℂ[x], but where the role of the variable x is replaced by a prime p, and the roles of a function f(x) and its derivative df/dx are now played by an integer α ∈ ℤ and its Fermat quotient δpa = (a - ap)/p.

In making our presentation of these type of operators, we find no better way than discussing the p-adic numbers in detail also, and some of the classical differential analysis on the field of p-adic numbers, emphasizing the aspects that give rise to the philosophy behind the arithmetic differential operators. The reader is urged to contrast these ideas at will, while keeping in mind that our study is neither exhaustive nor intended to be so, and most of the time we shall content ourselves by explaining the differential aspect of an arithmetic operator by way of analogy, rather than appealing to the language of jet spaces. But even then, the importance of these operators will be justified by their significant appearance in number theoretic considerations. One of our goals will be to illustrate how different these operators are when the ground field where they are defined is rather coarse, as are the p-adic integers ℤp that we use.