Preview

PDF, English Download (760kB) | Terms of use

Abstract

Let K be a number field, normal over Q, and p be an unramified prime in K|Q. We study s-sequences and analytic properties of their generating functions (which are s-fold derivatives of s-functions) where s refers to a natural number. The entries of such an s-sequence (a n )_n∈N are p-adic integral numbers and satisfy certain supercongruence relations that depend on s and involve the Frobenius element at every prime ideal dividing p. While the case s = 1 is widely studied in the literature, we are interested in the situation s ≥ 2. The obstruction of being an s-sequence grows for growing s. The first result in the present work is the statement that if the generating function of a 2-sequence represents a rational function, then the coefficients a n belong to a cyclotomic field. More precisely, we show that the poles of such functions are poles of order one given by roots of unity and rational residue. In the second part, we analyze an operator on formal power series, called framing, which preserves 2-functions. As a second result, we show that the image of rational 2-functions under the framing can be integrated to 3-functions, at least for almost all primes p. As a trivial consequence of this second theorem, we obtain the Jacobsthal-Kazandzidis congruence.