On -Difference Riccati Equations and Second-Order Linear -Difference Equations

We consider -difference Riccati equations and second-order linear -difference equations in the complex plane. We present some basic properties, such as the transformations between these two equations, the representations and the value distribution of meromorphic solutions of -difference Riccati equations, and the -Casorati determinant of meromorphic solutions of second-order linear -difference equations. In particular, we find that the meromorphic solutions of these two equations are concerned with the -Gamma function when such that . Some examples are also listed to illustrate our results. 1. Introduction and Main Results In this paper, a meromorphic function means meromorphic in the whole complex plane , unless stated otherwise. We also assume that the reader is familiar with the standard symbols and fundamental results such as , and , of Nevanlinna theory, see, for example, [1, 2], for a given meromorphic function . A meromorphic function is said to be a small function relative to if , where is used to denote any quantity satisfying as , possibly outside of a set of finite logarithmic measure, furthermore, possibly outside of a set of logarithmic density logdens . For a small function relative to , we define Recently, Ishizaki [3] considered difference Riccati equation and second-order linear difference equation where is meromorphic function, and gave surveys of basic properties of (2) and (3), which are analogues in the differential cases. Now, we are concerned with -difference Riccati equation and second-order linear -difference equation where , , and are rational functions and will obtain some parallel results for -difference case. For a meromorphic function , the -difference operator is defined by . This paper is organized as follows. In Section 2, we describe the transformation between -difference Riccati equation (4) and second-order linear -difference equation (5). In Section 3, we present some properties of -difference Riccati equation (4), such as -difference analogue on the property of a cross ratio for four distinct meromorphic solutions of a differential Riccati equation, the meromorphic solutions concerning with -Gamma function. In Section 4, we study the value distribution of transcendental meromorphic solutions of -difference Riccati equation (4) and the form of meromorphic solutions of second-order linear -difference equation (5). In Section 5, we discuss the properties on the -Casorati determinant of meromorphic solutions of second-order linear -difference equation (5). 2. Transformations between -Difference Riccati Equations and