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

Abstract

Let ff 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 cc are complex constants. The authors prove that Dq,cf(z)−af(z)−a=aa−β,\frac{{{\mathfrak{D}}}_{q,c}f\left(z)-a}{f\left(z)-a}=\frac{a}{a-\beta }, if Dq,cf(z){{\mathfrak{D}}}_{q,c}f\left(z) and f(z)f\left(z) share value a(≠β)a\left(\ne \beta ) CM, where Dq,cf(z)=f(qz+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].

Keywords

• hahn difference operator
• jackson q-difference operator
• picard exceptional values
• uniqueness
• 30d30