Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value

Abstract

Abstract Let f f be a transcendental entire function of finite order with a Picard exceptional value β ∈ C \beta \in {\mathbb{C}} , q ∈ C \ { 0 , 1 } q\in {\mathbb{C}}\setminus \left\{0,1\right\} and c c are complex constants. The authors prove that D q , c f ( z ) − a f ( z ) − a = a a − β , \frac{{{\mathfrak{D}}}_{q,c}f\left(z)-a}{f\left(z)-a}=\frac{a}{a-\beta }, if D q , c f ( z ) {{\mathfrak{D}}}_{q,c}f\left(z) and f ( z ) f\left(z) share value a ( ≠ β ) a\left(\ne \beta ) CM, where D q , c f ( z ) = f ( q z + c ) − f ( z ) ( q − 1 ) z + c {{\mathfrak{D}}}_{q,c}f\left(z)=\frac{f\left(qz+c)-f\left(z)}{\left(q-1)z+c} is the Hahn difference operator. This result generalizes the related results of Zongxuan Chen [On the difference counterpart of Brück’s conjecture, Acta Math. Sci. 34B (2014), 653–659].