The paper deals with characteristic funtion and deficiency of a meromorphicfunction. We mainly focused on the relation between the characteristic function ofa product of difference operators with the characteristic function of a meromorphicfunction with maximal deficiency sum. The concept of maximal deficiency sum ofa meromorphic function is employed as an effective tool for our research. In the samecontext, the notion of a difference polynomial of a difference operator is discussed.The paper contains the details analysis and discussion of some asymptotic behaviourof 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 differencepolynomial of difference operators. Several innovative idea to establish some inequalitieson 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 onproduct and polynomial of difference operators of our article improved andgeneralised the results of Z. Wu.

In this paper, we shall study the uniqueness problems of differential polynomials of meromorphic functions sharing 1 value. Here we prove two uniqueness theorems which extend and improve recent results of H.P. Waghamore and N.H. Sannappala [10]. 1 corresponding author 2010 Mathematics Subject Classification. 30D35. 119 120 N. MANDAL AND A. SHAWm ≤ l and l + 1 times if m > l. If E l (a, f ) = E l (a, g), then we say that f and g share the value a with weight l. We also use the notation N (r, a; f |p) to denote the counting function of f − a where m fold zeros is counted m times if m ≤ p and p times if m > p , where p is an integer. Definition 1.1. [4] Let f is a nonconstant meromorphic function and a ∈ C ∪ {∞}, the counting function of a-points of f with multiplicities at least p(∈ Z + ) is denoted by N (r, a; f |≥ p) and N (r, a; f |≥ p) is the corresponding reduced counting function. Similarly we can define N (r, a; f |≤ p) and N (r, a; f |≤ p). Definition 1.2. [4] The counting function of a-points of f , where an a-point of multiplicities m is counted m times if m ≤ p and p times if m > p is denoted by N p (r, a; f ), where p ∈ Z + ∪ {∞}. Then we can write: