On some properties of solutions of certain q-difference equations in ultrametric fields

Abstract

Let K be a complete, algebraically closed ultrametric field, and let ℳ(K) denote the field of meromorphic functions over K. In this article, we investigate the properties of solutions to certain ultrametric q-difference equations. Specifically, we examine the growth behavior of ultrametric meromorphic functions f that satisfy these equations. Furthermore, we establish necessary conditions on the coefficients {for} which a difference equation of the form:

∑^s_i=0 A_i(x)f(x+i) = B(x)

where B(x), A₀(x), …, A_s(x) (with s ≥ 1) {is a polynomial, admits a meromorphic solution.