Some Properties on Complex Functional Difference Equations

Abstract EN

We obtain some results on the transcendental meromorphic solutions of complex functional difference equations of the form ∑ λ ∈ I α λ ( z ) ( ∏ j = 0 n f ( z + c j ) λ j ) = R ( z , f ∘ p ) = ( ( a 0 ( z ) + a 1 ( z ) ( f ∘ p ) + ⋯ + a s ( z ) ( f ∘ p ) s ) / ( b 0 ( z ) + b 1 ( z ) ( f ∘ p ) + ⋯ + b t ( z ) ( f ∘ p ) t ) ) , where I is a finite set of multi-indexes λ = ( λ 0 , λ 1 , … , λ n ) , c 0 = 0 , c j ∈ ℂ ∖ { 0 } ( j = 1,2 , … , n ) are distinct complex constants, p ( z ) is a polynomial, and α λ ( z ) ( λ ∈ I ) , a i ( z ) ( i = 0,1 , … , s ) , and b j ( z ) ( j = 0,1 , … , t ) are small meromorphic functions relative to f ( z ) .

We further investigate the above functional difference equation which has special type if its solution has Borel exceptional zero and pole.