On some ultrametric q-difference equations

Let K be a complete ultrametric algebraically closed field and let M(K) be the field of meromorphic functions in all K. Let B (X), A(0)(X),..., A(s) (X) (s >= 1) be elements of K (X) such that A(0)(X)A(s)(X) not equal 0. This paper is aimed to study functions f is an element of M(K) which are solutions of the functional equation: Sigma(s)(i=0) A(i)(x)(sigma(i)(q)f)(x) = B(x), where q is an element of K, 0 < vertical bar q vertical bar < 1 and (sigma(q) f)(x) = f (qx). First we show that, if A(0)(X),..., A(s) (X), B(X) are constant, then f is a rational function. Next, we examine solutions of the above equation in the general case and give some characterizations of the order of growth of these solutions. (C) 2010 Elsevier Masson SAS. All rights reserved.