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Admissible Solutions of the Schwarzian Type Difference Equation
This paper is to investigate the Schwarzian type difference equation where is a rational function in with polynomial coefficients, , respectively are two irreducible polynomials in of degree , respectively . Relationship between and is studied for some special case. Denote . Let be an admissible solution of such that ; then for (≥2) distinct complex constants , In particular, if , then
1. Introduction and Results
Throughout this paper, a meromorphic function always means being meromorphic in the whole complex plane, and always means a nonzero constant. For a meromorphic function , we define its shift by and define its difference operators by In particular, for the case . We use standard notations of the Nevanlinna theory of meromorphic functions such as , , and and as stated in [1–3]. For a constant , we define the Nevanlinna deficiency by
Recently, numbers of papers (see, e.g., [4–12]) are devoted to considering the complex difference equations and difference analogues of Nevanlinna theory. Due to some idea of , we consider the admissible solution of the Schwarzian type difference equation: where is a rational function in with polynomial coefficients, , respectively , are two irreducible polynomials in of degree , respectively, . Here and in the following, “admissible” always means “transcendental.” And we denote from now on. For the existence of solutions of (3), we give some examples below.
Examples. (1) is an admissible solution of the Schwarzian type difference equation:
(2) is an admissible solution of the Schwarzian type difference equation
(3) Let , then solves the Schwarzian type difference equation: This example shows that (3) may admit polynomial solutions.
Considering the relationship between and in those examples above, we prove the following result.
Theorem 1. For the Schwarzian type difference equation (3) with polynomial coefficients, note the following.(i)If it admits an admissible solution such that , then In particular, if , then .(ii)If its coefficients are all constants and it admits a polynomial solution with degree , then and .
Remark 2. From examples (1) and (2), we conjecture that in Theorem 1(i). However, we cannot prove it currently. From example (3) given before, we see that the restriction on the coefficients in Theorem 1(ii) cannot be omitted.
For the Schwarzian differential equation, where , , and are as stated before; Ishizaki  proved the following result (see also Theorem in ).
For the Schwarzian type difference equation (3), we prove the following result.
Theorem 3. Let be an admissible solution of (3) with polynomial coefficients such that , and let be ≥2 distinct complex constants. Then In particular, if , then
Remark 4. From Theorem 1, under the condition in Theorem 3, we have in (11). The behavior of the zeros and the poles of in is essentially different from that in the . We wonder whether the restriction can be omitted or not.
The following lemma plays a very important role in the theory of complex differential equations and difference equations. It can be found in Mohon’ko  and Valiron  (see also Theorem in the book of Laine and Yang ).
Lemma 6 (see ). Let be a meromorphic function of hyper order , and . Then possibly outside of a set of with finite logarithmic measure.
Lemma 7 (see ). Let be a meromorphic function of hyper order , and . Then possibly outside of a set of with finite logarithmic measure.
From Lemma 7, we can easily get the following conclusion.
Lemma 8. Let be a meromorphic function of hyper order , and . Then possibly outside of a set of with finite logarithmic measure.
Lemma 9. Let be an admissible solution of (3) with coefficients. Then, using the notation , In particular, if , then
Proof. We use the idea by Ishizaki  (see also ) to prove Lemma 9. It follows from Lemma 8 that
From this and Lemma 5, we get
If , since all coefficients of and are polynomials, there are at the most finitely many poles of , neither the poles of nor the zeros of . Therefore, we see that We obtain (18) from this and (22) immediately.
If , there are at most finitely many poles of , not the zeros of , then Now (18) follows from (22) and (24).
Notice that if , then (24) always holds. This finishes the proof of Lemma 9.
3. Proof of Theorem 1
Case 1. Equation (3) admits an admissible solution such that . Since all coefficients of and are polynomials, there are at the most finitely many poles of that are not the poles of and . This implies that
From Lemma 5, we get
If , then , which yields that . That is a contradiction to our assumption. Thus, .
If , then , , and . Now from (3), we get Considering degrees of both sides of the equation above, we can see that .
If , we can deduce similarly that where are polynomials such that .
Rewrite (3) as follows:
From (34), we find that the leading coefficient of is
Considering degrees of both sides of (35), we prove that .
4. Proof of Theorem 3
Firstly, we consider the general case. As mentioned in Remark 1 in , due to Jank and Volkmann , if (3) admits an admissible solution, then there are at most common zeros of and . Since all coefficients of are polynomials, there are at the most finitely many poles of that are the zeros of . Therefore, from (3), we have Combining this and Lemma 9, applying the second main theorem, we get Thus, we prove that (10) holds.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the referees for their valuable suggestions. This work is supported by the NNSFC (nos. 11226091 and 11301091), the Guangdong Natural Science Foundation (no. S2013040014347), and the Foundation for Distinguished Young Talents in Higher Education of Guangdong (no. 2013LYM_0037).
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Copyright © 2014 Baoqin Chen and Sheng Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.