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), A0(X),…,As(X) (s⩾1) be elements of K(X) such that A0(X)As(X)≠0.This paper is aimed to study functions f∈M(K) which are solutions of the functional equation: , where q∈K, 0<|q|<1 and (σqf)(x)=f(qx).First we show that, if A0(X),…,As(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.