Derivative of power series attached to Gamma-transform of p-adic measures

Abstract

We extend the result of Angles (2007) [1], namely f'(T; theta) not equivalent to 0 (mod p) for the Iwasawa power series f(T; theta) is an element of (Z) over bar (p)parallel to T - 1 parallel to. For the derivative D = Td/dT, a numerical polynomial Q on Z(p), and a prime pi in (Z) over bar (p) over p, we show that Q(D)f(T: theta) equivalent to 0 (mod pi) if and only if Q equivalent to 0 (mod pi) i.e. Q (x) equivalent to 0 (mod pi) for all x is an element of Z(p). This result comes from a similar assertion for the power series attached to the Gamma-transform of a p-adic measure which is related to a certain rational function in (Z) over bar (p)parallel to T - 1 parallel to. (C) 2009 Elsevier Inc. All rights reserved