The class of meromorphic functions sharing values with their difference polynomials

The field of c-periodic meromorphic functions in $$ \mathbb {C} $$ C is defined by $$ {\mathcal {M}}_c:=\{f : f\; \text{ is } \text{ meromorphic } \text{ in }\; \mathbb {C}\;\text{ and }\; f(z+c)=f(z)\} $$ M c : = { f : f is meromorphic in C and f ( z + c ) = f ( z ) } and the c-shift linear difference polynomial of a meromorphic function f is defined by $$\begin{aligned} L^n_c(f)=a_nf(z+nc)+\cdots +a_1f(z+c)+a_0f(z), \end{aligned}$$ L c n ( f ) = a n f ( z + n c ) + ⋯ + a 1 f ( z + c ) + a 0 f ( z ) , where $$ a_n(\ne 0), \ldots , a_1, a_0\in \mathbb {C} $$ a n ( ≠ 0 ) , … , a 1 , a 0 ∈ C . It is easy to see that if $$ a_j=\left( {\begin{array}{c}n\\ j\end{array}}\right) (-1)^{n-j} $$ a j = n j ( - 1 ) n - j , then $$ L^n_c(f)=\Delta ^n_cf $$ L c n ( f ) = Δ c n f , where $$ \Delta ^n_cf $$ Δ c n f is a higher difference operator of f. Let $$\begin{aligned} {\mathcal {S}}_c=\{f: f \; \text{ is } \text{ meromorphic } \text{ in } \; \mathbb {C}\;\text{ and }\; L^n_c(f)\equiv f\}. \end{aligned}$$ S c = { f : f is meromorphic in C and L c n ( f ) ≡ f } . In this paper, we study the value sharing problem between a meromorphic functions f and their linear difference polynomials $$ L^n_c(f) $$ L c n ( f ) and prove a result generalizing several existing results. In addition, we find the class $$ {\mathcal {S}}_c $$ S c completely which gives the positive answers to a conjecture and an open problem in this direction.