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

Abstract Let f f be a transcendental entire function of finite order with Picard exceptional value β ∈ C \beta \in {\mathbb{C}} , q \ { 0 , 1 } q\in {\mathbb{C}}\setminus \left\{0,1\right\} and c c are complex constants. The authors prove that xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> mathvariant="fraktur">D ( z ) − a = \frac{{{\mathfrak{D}}}_{q,c}f\left(z)-a}{f\left(z)-a}=\frac{a}{a-\beta }, if {{\mathfrak{D}}}_{q,c}f\left(z) f\left(z) share ≠ a\left(\ne ) CM, where + {{\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 related results Zongxuan Chen [ On counterpart Brück’s conjecture Acta Math. Sci. 34B (2014), 653–659].