On meromorphic function with maximal deficiency sum and it's difference operators

The paper deals with characteristic funtion and deficiency of a meromorphic function. We mainly focused on the relation between the characteristic function of a product of difference operators with the characteristic function of a meromorphic function with maximal deficiency sum. The concept of maximal deficiency sum of a meromorphic function is employed as an effective tool for our research. In the same context, the notion of a difference polynomial of a difference operator is discussed. The paper contains the details analysis and discussion of some asymptotic behaviour of the product of difference operators, such as $\lim_{r\rightarrow \infty }\frac{T(r,\prod_{i=1}^{q}\Delta _{\eta_{i}}f)}{T(r,f)}$, $\lim_{r\rightarrow \infty }\frac{N(r,0;\prod_{i=1}^{q}\Delta_{\eta_{i}}f)} {T(r,\prod_{i=1}^{q}\Delta _{\eta _{i}}f)}$, $\overline{\lim}_{r\rightarrow\infty}\frac{N(r,\infty; \prod_{i=1}^{q}\Delta_{\eta_{i}}f)+N(r,0;\prod_{i=1}^{q} \Delta_{\eta_{i}}f)}{T(r,\prod_{i=1}^{q}\Delta_{\eta_{i}}f)}$ etc. and same resolution and discussion also developed for the difference polynomial of difference operators. Several innovative idea to establish some inequalities on the zeros and poles for $\prod_{i=1}^{q}\Delta _{\eta _{i}}f$ and $L(\Delta_{\eta}f)$ are also introduced. We broadly elaborate our results with many remarks and corollaries, and give two excellent examples for proper justification of our results. The results on product and polynomial of difference operators of our article improved and generalised the results of Z. Wu.