As is well known, in the twentieth century, the theories of many mathematical embranchments obtained speedy development, while the theory of meromorphic functions (Nevanlinna distribution theory) also obtained a speedy development; there is a vast and comprehensive literature on the derivatives of meromorphic functions and the complex differential equations. But the literature on analytic difference of meromorphic functions and the complex difference equations is very scarce and scattered.

In more recent years, activity in the area of the complex differences and the complex difference equations has fleetly increased. Among pioneering work in the area we mention in particular is a recent paper by Ablowitz et al. (Nonlinearity, 13(2000), 889-905). Their starting point is the observation that all of the pertinent complex difference equations admit obvious analytic solutions and hence can be studied in the method of complex analysis. Subsequently, Halburd and Korhonen (Journal of Mathematical Analysis and Applications, 314(2006), 477-487) and Chiang and Feng (Ramanujan Journal, 16(2008), 105-129), respectively, obtained the difference analogue of logarithm derivative of meromorphic function which is a particularly useful tool for investigation of complex differences. The complex difference equations are of important roles in physics, chemistry, quantum mechanics, and economics. The theory of the complex differences and the complex difference equations should obtain a speedy development.

We invite investigators to contribute original research articles as well as review articles that will stimulate the continuing efforts to the complex differences and the complex difference equations. Submissions for this special issue should be of the following list of potential topics with high current interest falling within the scope of Abstract and Applied Analysis. Potential topics include, but are not limited to:

• Difference analogues of Nevanlinna distribution theory
• Properties of difference polynomials
• Distribution properties of meromorphic solutions of difference equations, such as linear difference equations, difference Painleve equations, and difference Riccati equations
• Existence of meromorphic solutions of difference equations
• Q-difference analogues of Nevanlinna distribution theory
• Distribution properties of meromorphic solutions of q-difference equations
• Unicity of meromorphic functions concerning their differences and shifts
• Related topics with difference such as the complex differential equations, discrete Painleve equations, and other integrable discrete systems in one independent variable