Growth of Meromorphic Solutions of Some q-Difference Equations

and Applied Analysis 3 where |q| > 1 and the index set J consists of m elements and the coefficients a i (z) (a n (z) = 1) and b J (z) are small functions of f. If f is of finite order, then |q| < n + m − 1. 2. Some Lemmas The following important result by Valiron andMohon’ko will be used frequently, one can find the proof in Laine’s book [16, page 29]. Lemma 9. Let f be a meromorphic function. Then, for all irreducible rational function in f,


Introduction and Main Results
In this paper, we mainly use the basic notation of Nevanlinna Theory, such as (, ), (, ), and (, ), and the notation (, ) is defined to be any quantity satisfying (, ) = ((, )) as  → ∞ possibly outside a set of  of finite linear measure (see [1][2][3]).In addition, we use the notation () to denote the order of growth of the meromorphic function () and () to denote the exponent of convergence of the zeros.We also use the notation () to denote the exponent of convergence of fixed points of .We give the definition of () as following.
In particular, we concern the second-order -difference equation with rational coefficients, that is, in the case of  = 2. From Theorem 2, we know that if  is a nonconstant meromorphic solution, then  ≤ 2. Thus, the second-order -difference equation is the following form: First of all, we give some remarks.
Remark 3. If  2 () and  2 () are not zero at the same time, by Theorem 2, we derive that the solution of ( 4) is of order zero.
Remark 5.If  2 () =  2 () = 0,  1 () ̸ = 0, by Theorem 2, the order of the solutions is less than log(2)/(− log ||).Thus, a question arises: does the equation have a solution which is of order nonzero under this situation?This question is still open.
In [6], Chen and Shon proved some theorems about the properties of solutions of the difference Painlevé I and II equations, such as the exponents of convergence of fixed points and the zeros of transcendental solutions.A natural question arises: how about the exponents of convergence of the fixed points and the zeros of transcendental solutions of the -difference equation (4)?Do the transcendental solutions have infinitely many fixed points and zeros?The following theorem, in which the coefficients are constants, answers the above questions partly.Theorem 6. Suppose that  is a transcendental solution of the equation where || < 1, the coefficients  1 ,  0 ,  1 ,  2 ,  0 ,  1 , and  2 are constants, and at least one of  2 ,  2 is nonzero.Then, () = 0 and (i) has infinitely many fixed points, and (ii) has infinitely many zeros, whenever  0 ̸ = 0.
In the rest of the paper, we consider (3) when || > 1.In [15], Heittokangas et al. considered the essential growth problem for transcendental meromorphic solutions of complex difference equations, which is to find lower bounds for their characteristic functions.Following this idea, Zheng and Chen [14] obtained the following theorem for -difference equations.

Theorem C. Suppose that 𝑓 is a transcendental solution of equation
where  ∈ C, || > 1, the coefficients   () are rational functions, and ,  are relatively prime polynomials in  over the field of rational functions satisfying  = deg  ,  = deg  , and  =  −  ≥ 2. If  has infinitely many poles, then for sufficiently large , (, ) ≥  log /( log ||) holds for some constant  > 0. Thus, the lower order of , which has infinitely many poles, satisfies () ≥ log /( log ||).
Regarding Theorem C, they obtained the lower bound of the order of solutions.Then, how about the upper bound of the order of the solutions?Can the conditions of Theorem C become a little more simple?In fact, we have the following theorem.
We know that the difference analogues and -difference analogues of Nevanlinna's theory have been investigated.Consequently, many results on the complex difference equations and -difference equations have been obtained respectively.Thus, mixing the difference and -difference equations together is a natural idea.The following Theorem 8 is just a simple application of the above idea, and further investigation is required.
In what follows, we will consider difference products and difference polynomials.By a difference product, we mean a difference monomial, that is, an expression of type where  1 , . . .,   are complex numbers and  1 , . . .,   are natural numbers.A difference polynomial is a finite sum of difference products, that is, an expression of the form where   ( ∈ ) is a set of distinct complex numbers and the coefficients   () of difference polynomials are small functions as understood in the usual the Nevanlinna theory; that is, their characteristic is of type (, ).
Theorem 8. Suppose that  is a nonconstant meromorphic solution of the equation where || > 1 and the index set  consists of  elements and the coefficients   () (  () = 1) and   () are small functions of .If  is of finite order, then || <  +  − 1.
The next lemma on the relationship between (, ()) and (||, ()) is due to Bergweiler et on a set of logarithmic density 1.