Meromorphic solutions of certain type of differential-difference equations

Abstract

We consider meromorphic solutions to general differential-difference equation of type L ( z , f ) + ∑ j = 1 q c j ( z ) f n j ( z + ζ j ) = ∑ j = 1 r γ j e λ j z , where L ( z , f ) ≢ 0 is a linear differential-difference polynomial of f with small function coefficients, and 1 < n 1 < ⋯ < n q , γ j ( 1 ≤ j ≤ r ) are nonvanishing constants, c j ( z ) ( 1 ≤ j ≤ q ) are nonvanishing small functions. If the equation admits a transcendental meromorphic solution f of finite order satisfying that λ ( f ) < σ ( f ) , λ ( 1 / f ) < σ ( f ) , then representation of f and relations between coefficients in ψ ( z , f ) and γ i , λ j can be obtained.