Complete Classification of Meromorphic Solutions of a Fermat-Type Difference Equation

We investigate meromorphic solutions of the difference equation f ( z ) 2 + f ( z + c ) = h ( z ) , c ∈ C ∖ { 0 } , align* f(z)²+f(z+c)=h(z), c∈ C\0\, align* where h is a transcendental meromorphic function satisfying N ( r , h ) + N ( r , 1 / h ) = S ( r , h ) . N(r,h)+N(r,1/h)=S(r,h). upper N left parenthesis r comma h right parenthesis plus upper N left parenthesis r comma 1 divided by h right parenthesis equals upper S left parenthesis r comma h right parenthesis period We prove that every solution of hyper-order less than 1 $1$ 1 is of the form f ( z ) = e a z + b − 1 , h ( z ) = e 2 a z + 2 b , align* f(z)=eᵃᶻ⁺ᵇ-1, h(z)=e²ᵃᶻ⁺²ᵇ, align*