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Journal of Complex Analysis
Volume 2013 (2013), Article ID 938579, 10 pages
http://dx.doi.org/10.1155/2013/938579
Research Article

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

1School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
2Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, Finland

Received 29 August 2012; Accepted 17 October 2012

Academic Editor: Yan Xu

Copyright © 2013 Zhi-Bo Huang. 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.

Abstract

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 Linear -Difference Equations of Second-Order

It is well known that a differential Riccati equation and second-order linear differential equation are closely related by the transformation where is a meromorphic function, see, for example, [4, pages 103–106].

Ishizaki [3] considered a difference analogue of (6) and (7) and obtained that difference Riccati equation (2) and second-order linear difference equation (3) are closely linked by the transformation where is a meromorphic function.

Here, we are concerned with a transformation between (4) and (5), see [5]. For a nontrivial meromorphic solution of (5), we take Then satisfies -difference Riccati equation (4). In fact, we deduce from (5) that which implies the desired form of (4).

Conversely, if (4) admits a nontrivial meromorphic solution , then meromorphic function of first-order -difference equation (10) satisfies (5). In fact, we conclude from (4) and (10) that which implies (5).

Example 1. Suppose that and . Let and . Then and satisfy -difference Riccati equation (4) and second-order linear -difference equation (5), respectively, which both satisfy the transformation (10).

3. Representations of Solutions of -Difference Riccati Equations

The representations on meromorphic solutions of Riccati equations are interesting. Bank et al. [6, pages 371–373] obtained that differential Riccati equation (6) possesses a one parameter family of meromorphic solutions if (6) has three distinct meromorphic solutions , and . Ishizaki extended this property to difference Riccati equation (2) and obtained a difference analogue of this property, see [3, Proposition 2.1]. Now, we present this property for -difference case below, which can also be seen as a -difference analogue of the fact that a cross ratio for four distinct meromorphic solutions of a differential Riccati equation is a constant, see, for example, [4, pages 108-109]. Furthermore, we find that meromorphic solutions of -difference Riccati equations (4) are concerned with - Gamma function if such that .

Theorem 2. Suppose that (4) possesses three distinct meromorphic solutions , and . Then any meromorphic solution of (4) can be represented by where is a meromorphic function satisfying . Conversely, if for any meromorphic function satisfying , we define a function by (13), then is a meromorphic solution of (4).

Proof of Theorem 2. Let be distinct meromorphic functions. We denote a cross ratio of by Suppose that is meromorphic solution of (4) and is also distinct from , and . We first show that is a meromorphic solution of -difference Riccati equation (4) if and only if , where . In fact, we conclude from (4) that Conversely, if , then We conclude from (16) that , which shows that satisfies (4).
Thus, for any meromorphic function satisfying , we define by Then is represented by (13), and also satisfies -difference Riccati equation (4). The proof of Theorem 2 is completed.

Now, we recall some results of transcendental meromorphic solutions concerned with -difference Riccati equation (4). Bergweiler et al. [7, 8] pointed out that all transcendental meromorphic solutions of (5) satisfy if and . Since (10) is a transformation between (4) and (5), we obtain that all transcendental meromorphic solutions of (4) are of order zero if and . On the other hand, if is a transcendental meromorphic solution of where and the coefficients of are small functions relative to , Gundersen et al. [9] showed that the order of growth of (18) is equal to , where is the degree of irreducible rational function in . Thus, from the above two cases, we obtain that all transcendental meromorphic solutions of (4) are of order zero for all and .

We also illustrate some of the results on -difference equations, which are explicitly solvable in terms of known zero-order meromorphic functions (see [5]). Let be such that . Then -Gamma function is defined by where . It is a meromorphic function with poles at , where and are nonnegative integers, see [10]. By defining and , we see that is a meromorphic function of zero-order with no zeros, having its poles at .

Therefore, the first-order linear -difference equation is solved by the function . Moreover, for general first-order linear -difference equation, where is a rational function. If is a constant, (22) is solvable in terms of rational functions if and only if is an integer. If is nonconstant, let and be the zeros and poles of , respectively, repeated according to their multiplicities. Then can be written in the form where is a complex number depending on . So, (22) is solved by which is meromorphic if and only if is an integer.

Now, let and be two distinct rational solutions of the differential Riccati equation (6). If there exists a rational solution distinct from , then all meromorphic solutions of (6) are rational solutions. If there exists a transcendental meromorphic solution , then there is no rational solution other than , see, for example, [6, pages 393-394]. For difference Riccati equation (2), Ishizaki obtained a difference analogue, see [3, Proposition 2.2]. In the following, we give a -difference case for -difference Riccati equation (4).

Theorem 3. Let be such that . Suppose that -difference Riccati equation (4) possesses two distinct rational solutions and . Then there exists a meromorphic solution distinct from and so that any meromorphic solution of (4) is represented in the form (13).

Proof of Theorem 3. Since and are two distinct rational solutions of (4), we define a translation Then . Substituting (25) into (4), we conclude that which is type of (22). So, is a meromorphic solution of (26) as in the form (24). Therefore, we conclude from (25) that is a meromorphic solution of (4), which is distinct from and . So, we now deduce from Theorem 2 that any meromorphic solution of (4) is represented in the form (13). The proof of Theorem 3 is completed.

Example 4. Let , and in (4) and (5). Then functions satisfy -difference Riccati equation (4), and (26) turns into We note that
Thus, we conclude from (24) and (29) that is a meromorphic solution of (4), which is distinct from and . Moreover, we also conclude from (10), (27), and (5) that which are corresponding to and , respectively, and are also the types of (22). Thus, we deduce from (24) that satisfy second-order linear -difference equation (5).

4. Value Distribution of Solutions of -Difference Riccati Equations and Form of Solutions of Second-Order Linear -Difference Equations

We first consider the value distribution of transcendental meromorphic solution of -difference Riccati equation (4).

Theorem 5. Let and be nonconstant rational functions. If is a zero-order transcendental meromorphic solution of -difference Riccati equation with and , then (i)if then has at most one Borel exceptional value;(ii)if , then Nevanlinna deficiencies ;(iii)if and , then has infinitely many fixed points.

In particular, we obtain the following theorem.

Theorem 6. If and are constants, and if and , then -difference Riccati equation (4) has only rational solutions. Furthermore, if and is nonzero constant, then (4) has only a nonzero constant solution , which satisfies .

We need some preliminaries to prove Theorems 5 and 6.

The theorem of Tumura and Clunie is an important result in Nevanlinna theory, see [11, 12]. Weissenborn extended it and obtained the following lemma.

Lemma 7 (see [13, Theorem]). Let be a meromorphic function and let be given by Then either or

Lemma 8. Suppose that is a nonconstant meromorphic function satisfying Let be a polynomial in with , and coefficients satisfying Then or Thus, .

Proof of Lemma 8. By differentiating both sides of (39), we conclude that Thus, we deduce from (39) and (43) that Therefore, is a polynomial in with degree no greater than and the term of degree zero is . Then Otherwise, if , then is a nonzero constant, a contradiction. We also note that is a small function relative to by (38) and the lemma of logarithmic derivative. Set Then and are small functions relative to and is a polynomial in with degree no greater than and the term of degree zero is small function relative to .
If the degree of is greater than zero, then by repeating the above process, we can get two small functions and such that is a polynomial in with a degree less than the degree of and the term of degree zero is a small function relative to .
We note that such process will be terminated at most times. Thus, We can proceed this process to obtain small functions and , where and , such that are polynomial in with , where and is a small function relative to . Thus, we deduce that the small function can be expressed as a linear differential polynomial in with coefficients being small functions relative to . So, On the other hand, we deduce from Lemma 7 that either or Thus, we deduce from Valiron-Mohon’ko Lemma, (51), and (52) that (41) holds and obtain from (38) and (53) that (42) holds. Therefore, . The proof of Lemma 8 is completed.

Lemma 9 (see [9, Theorem 5.2]). Let be a transcendental meromorphic solution of with meromorphic coefficients relative to and such that . If , then (54) is either of the form

Lemma 10 (see [5, Theorem 2.2]). Let be a nonconstant zero-order meromorphic solution of where is a -difference polynomials in . If for a small function relative to , then on a set of logarithmic density 1.

Lemma 11 (see [2, Theorem and Corollary ]). Let be a meromorphic function. Then for all irreducible rational functions in , with meromorphic coefficients , the characteristic function of satisfies where and .
In the particular case when we have We also use the observation [7, page 2] that, for any meromorphic function and any constant ,

Proof of Theorem 5. Suppose that is a zero-order transcendental meromorphic solution of -difference Riccati equation (4).
(i) Suppose that has two finite Borel exceptional values and . For the case where one of and is infinite, we can use a similar method to prove. Set Since , we deduce from (63) that We also conclude from (63) that Now, substituting (65) into (4), we conclude that By the assumptions of Theorem 5, we get Thus, we deduce from Lemma 9, (64), and (66) that where .
If , we conclude from (66) and (68) that Thus, we deduce from Lemma 8 and (64) that (69) is a contradiction. If , we use the same method as above to get another contradiction. Therefore, at most one Borel exceptional value.
(ii) We first prove . We obtain from (4) that Since , we deduce from Lemma 10 and (70) that on a set of logarithmic density 1. Therefore, Thus, .
We second prove . Set . Then Now, substituting into (4), we conclude that Since , we obtain from Lemma 10 and (74) that on a set of logarithmic density 1. Therefore, on a set of logarithmic density 1. Thus, we conclude from and (76) that on a set of logarithmic density 1, and so, Thus, .
(iii) Set . Then Substituting into (4), we conclude that Since , we deduce from Lemma 10 and (80) that on a set of logarithmic density 1. Therefore on a set of logarithmic density 1. This shows that has infinitely many fixed points if .

Proof of Theorem 6. Suppose first that and (4) with nonzero constant coefficients and admits a meromorphic solution . We assert that is rational. In fact, we conclude from Lemma 11, (4), and (62) that where (>0) is fixed number.
Thus, for any , there exists an such that By an inductive argument, we deduce from (84) that Suppose now that and (4) with nonzero constant coefficients and admits a meromorphic solution . Replacing by in (4), we proceed in a similar method as above to get (85) again. Therefore, is rational solution of (4).
Now, we affirm that must be nonzero constant if and is a constant. Otherwise, if is nonconstant rational and has a pole , we conclude from (4) that has infinitely many poles of the forms and infinitely many zeros of the forms for all . Conversely, If is nonconstant rational and has a zero , we conclude from (4) that has infinitely many zeros of the forms and infinitely many poles of the forms for all . These are both impossible since is rational. Thus, the only possible pole (resp. zero) of is at . So may have the form , where is a nonzero constant. If , we get a contradiction from (4). Therefore, and (4) has only a nonzero constant solution , which satisfies . The proof of Theorem 6 is completed.

We now consider the form of meromorphic solutions of (5), which is according to Theorem 6. In fact, more details about meromorphic solutions of (5) have been studied in [7, 14]. Here, we only consider the case that all coefficients are constants.

Theorem 12. If and is constant, and if and , then every meromorphic solution of second-order linear -difference equation (5) has the form , where and satisfying .

We first list a lemma needed below.

Lemma 13 (see [14, Theorem 2.1]). Suppose that and . Let be complex constants and let be of the reduced form , where is a polynomial of degree and . Then all meromorphic solutions of are of the reduced form , where is a polynomial and .

Proof of Theorem 12. We deduce from Lemma 13 that all meromorphic solutions of (5) are of the form , where and are defined as Lemma 13. Thus, we conclude from Theorem 6 and (10) that where is defined as Theorem 6. From (87), we obtain that there exists and such that , and so , where . Now, substituting into (5), we conclude that satisfies . The proof of Theorem 12 is completed.

Example 14. Let , and . Then second-order -difference equation (5) is solved by . Obviously, and satisfy the conclusions described by Theorem 12.

5. Linear -Difference Equations of Second-Order

Let and be meromorphic solutions of (5). We define the -Casorati determinant of meromorphic functions and by Then the -Casorati determinant vanishes identically on if and only if the functions and are linearly dependent over the field of functions . On the other hand, and are linear independent if and only if . From this definition, we have some properties on the -Casorati determinant as follows.

Theorem 15. If and are nontrivial meromorphic solutions of (5), then -Casorati determinant satisfies a first-order -difference equation Conversely, we assume that and satisfy (89). If is a meromorphic solution of (5), then is also a meromorphic solution of (5).

Proof of Theorem 15. Suppose first that and are nontrivial meromorphic solutions of (5), we conclude that Therefore, Second, if and satisfy (89), then we have We note that, for any meromorphic function , In particular, we take . Thus, So, we have From this, we conclude that Since is a meromorphic solution of (5), we have and so, This shows that is a meromorphic solution of (5). The proof of Theorem 15 is completed.

Theorem 16. (i) Let and be linear independent meromorphic solutions of (5), and let be the -Casoratian determinant of and . Then is represented as , where satisfies
(ii) Let be a nontrivial meromorphic solution of (5), and let be a meromorphic solution of (89). If satisfies (99), then is a meromorphic solution of (5).

Proof of Theorem 16. (i) From the definition of , we obtain This shows that satisfies first-order -difference equation of type By substituting into (101), we conclude that and so we obtain the desired form (99).
(ii) Obviously, we conclude from (99) and (89) that Since , , and we conclude from (103), and (104) that This yields that is a meromorphic solution of (5). The proof of Theorem 16 is completed.

Acknowledgments

The author would like to thank the referees for their helpful remarks and suggestions to improve this paper. The author also wishes to express his thanks to Professors Risto Korhonen, Ilpo Laine, Benharrat Belaïdi, and Kuldeep Singh Charak for their valuable advice during the preparation of this paper. This research was supported by National Natural Science Foundation of China (no. 11171119 and no. 11171121).

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