Meromorphic solutions on the first order non-linear difference equations

Abstract

Steinmetz [16] considered the first order non-linear differential equations $$C(z, f)(f^\prime)^2+B(z, f)f^\prime+A(z, f)=0,$$ where A(z, f), B(z, f), C(z, f) are polynomials in f with rational coefficients in z and pointed out that the above equation must reduce into some certain types when it admits transcendental meromorphic solutions. In this paper, we will consider its difference version $$C(z, f)f(z+c)^2+B(z, f)f(z+c)+A(z,f)=0.$$ We explore the conditions when the above difference equation admits transcendental meromorphic (entire) solutions. In addition, the difference equations which are similar to Fermat difference equations also be selected out and considered.