SOME PROPERTIES OF MEROMORPHIC SOLUTIONS FOR q-difference EQUATIONS

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1 Electronic Journal of Differential Equations, Vol , No. 75, pp. 2. ISSN: URL: or SOME PROPERTIES OF MEROMORPHIC SOLUTIONS FOR q-difference EQUATIONS HONG YAN XU, SAN YANG LIU, XIU MIN ZHENG Abstract. The main purpose of this article is to investigate some properties on the meromorphic solutions of some types of q-difference equations, which can be seen the q-difference analogues of Painevé equations. We obtain estimates of the exponent of convergence of poles of qfz := fqz fz, which extends some earlier results by Chen et al.. Introduction and statement of main results Throughout this paper, the term meromorphic will mean meromorphic in the complex plane C. Also, we shall assume that readers are familiar with the fundamental results and the standard notation of the Nevanlinna value distribution theory of meromorphic functions such as mr, f, Nr, f, T r, f, etc. see Hayman [3], Yang [25] and Yi and Yang [26]. We use σf, f and /f to denote the order, the exponent of convergence of zeros and the exponent of convergence of poles of fz respectively, and we also use Sr, f denotes any quantity satisfying Sr, f = ot r, f for all r on a set F of logarithmic density, the logarithmic density of a set F is defined by lim sup r log r [,r] F t dt. Throughout this article, where the set F of logarithmic density will be not necessarily the same at each occurrence. A century ago, Painlevé and his colleagues [2] considered the class w z = F z; w; w, where F is rational in w and w and locally analytic in z. They singled out a list of 50 equations, six of which could not be integrated in terms of known functions. These equations are now known as the Painlevé equations. The first two of these equations are P I and P II : w = 6w 2 + z, w = 2w 2 + zw + α, where α is a constant. However, after that, essentially nothing happened until about 980, and just after that differential Painlevé equations became an important research subject. 200 Mathematics Subject Classification. 39A50, 30D35. Key words and phrases. Meromorphic function; q-difference equation; zero order. c 207 Texas State University. Submitted February 4, 207. Published July 0, 207.

2 2 H. Y. XU, S. Y. LIU, X. M. ZHENG EJDE-207/75 In the 990s, the discrete Painlevé equations have become important research problems see [6, 8]. For example, the following equations y n+ + y n = an + b + c, y n+ + y n = an + b + c y n y n yn 2, are some known as the special discretization of discrete P I, and the equation y n+ + y n = an + by n + c yn 2, is known as the special discretization of the discrete P II, where a, b, c are constants, n N. Recently, a number of papers see [5, 4, 7] focused on complex difference equations and difference analogues of Nevanlinn s theory. Around 2006s, Halburd and Korhonen [0,, 2] used Nevanlinna value distribution theory to single out the difference Painlevé I and II equations from the following form wz + + wz = Rz, w,. where Rz, w is rational in w and meromorphic in z. They obtained that if. has an admissible meromorphic solution of finite order, then either w satisfies a difference Riccati equation, or. can be transformed by a linear change in w to some difference equations, which include difference Painlevé I equations wz + + wz = az + b + c,.2 wz az + b wz + + wz = + c wz wz 2,.3 and difference Painlevé II equation wz + + wz = az + bwz + c wz 2..4 Chen et al [3, 4, 22] studied some properties of finite order transcendental meromorphic solutions of.2.4, and obtained a lot of interesting results. In 2007, Barnett, Halburd, Korhonen and Morgan [] firstly established an analogue of the Logarithmic Derivative Lemma on q-difference operators. Closely related to difference expressions are q-difference expressions, where the usual shift fz + c of a meromorphic function will be replaced by the q-difference fqz, q C\{0, }. By this way, there were lots of results about difference operators, difference equations, q-difference operators, q-difference equations, and so on see [7, 9, 8, 9, 20, 24, 27, 28, 29]. In 205, Qi and Yang [23] investigated the equations fqz + f z q = az + b fz + c,.5 fqz + f z az + bfz + c = q fz 2,.6 which can be seen q-difference analogues of.2 and.4, and obtained some theorems as follows. Theorem. [23, Theorem.]. Let fz be a transcendental meromorphic solution with zero order of.5, and a, b, c be three constants such that a, b cannot vanish simultaneously. Then

3 EJDE-207/75 SOME PROPERTIES OF MEROMORPHIC SOLUTIONS 3 i fz has infinitely many poles. ii If a 0 and any d C, then fz d has infinitely many zeros. iii If a = 0 and fz takes a finite value A finitely often, then A is a solution of 2z 2 cz b = 0. Theorem.2 [23, Theorem.3]. Let a, b, c be constants with ac 0, and let fz be a transcendental meromorphic solution with zero order of equation.6. Then fz has infinitely many poles and fz d has infinitely many zeros, where d C. In this article, we further investigated some properties of transcendental meromorphic solutions of the equations.5,.6 and fqz + f z q = az + b fz + c fz 2,.7 and obtained the following theorems, which extends the previous results given by Qi and Yang [23]. Theorem.3. Let a, b, c be constants with a + b 0. Suppose that fz is a zero order transcendental meromorphic solution of.5. Then i if a 0, pz is a polynomial of degree k 0 and q, then fz pz has infinitely many zeros and f p = σf; if a = 0, then Borel exceptional values of fz can only come from the set E = {z 2z 2 cz b = 0}; ii f = qf = σ qf = σf. Theorem.4. Let a, b, c be constants with a + b + c 0. Suppose that fz is a zero order transcendental meromorphic solution of.6. Then i if a 0, pz is a polynomial of degree k 0 and q, then fz pz has infinitely many zeros and f p = σf; if a = 0, then Borel exceptional values of fz can only come from the set E = {z 2z 3 + b 2z + c = 0}; if c 0, then f = σf; ii f = qf = σ qf = σf. Theorem.5. Let a, b, c be constants with a + b + c 0. Suppose that fz is a zero order transcendental meromorphic solution of.7. Then i if a = 0, then Borel exceptional values of fz can only come from the set E = {z 2z 3 bz c = 0}; ii f = qf = σ qf = σf. 2. Some Lemmas The following result can be called an analogue of q-difference Clunie lemma, recently proved by Barnett et al. [, Theorem 2.]. Here a q-difference polynomial of f for q C\{0, } is a polynomial in fz and finitely many of its q-shifts fqz,..., fq n z with meromorphic coefficients in the sense that their Nevanlinna characteristic functions are ot r, f on a set of logarithmic density. Lemma 2. [6, Theorem 2.5]. Let f be a transcendental meromorphic solution of order zero of a q-difference equation of the form U q z, fp q z, f = Q q z, f,

4 4 H. Y. XU, S. Y. LIU, X. M. ZHENG EJDE-207/75 where U q z, f, P q z, f and Q q z, f are q-difference polynomials such that the total degree deg U q z, f = n in fz and its q-shifts, whereas deg Q q z, f n. Moreover, we assume that U q z, f contains just one term of maximal total degree in fz and its q-shifts. Then on a set of logarithmic density. mr, P q z, f = ot r, f, Lemma 2.2 [, Theorem 2.5]. Let f be a nonconstant zero-order meromorphic solution of P q z, f = 0, where P q z, f is a q-difference polynomial in fz. If P q z, a 0 for slowly moving target az, then mr, on a set of logarithmic density. = ot r, f, f a Lemma 2.3 [27, Theorem. and.3]. Let fz be a nonconstant zero-order meromorphic function and q C \ {0}. Then T r, fqz = + ot r, fz, on a set of lower logarithmic density. Nr, fqz = + onr, fz, Lemma 2.4 Valiron-Mohon ko [5]. Let fz be a meromorphic function. Then for all irreducible rational functions in f, m i=0 Rz, fz = a izfz i n j=0 b jzfz, j with meromorphic coefficients a i z, b j z, the characteristic function of Rz, fz satisfies that T r, Rz, fz = dt r, f + OΨr, where d = max{m, n} and Ψr = max i,j {T r, a i, T r, b j }. 3. Proof of Theorem.3 Suppose that fz is a zero order transcendental meromorphic solution of.5. i a 0. Let pz is a polynomial of degree k and pz = a k z k Let gz = fz pz. Substituting fz = gz + pz into equation.5, we have It follows that Then, we have gqz + pqz + g z q + pz q = az + b gz + pz + c. P q z, g :=[gqz + pqz + g z q + pz ][gz + pz] q az + b c[gz + pz] = P q z, 0 = [pqz + p z ]pz az + b cpz. 3.2 q If pz 0, then P q z, 0 = az + b 0. If k = 0 and a 0 α C \ {0}, then P q z, 0 = 2α 2 az + b cα 0.

5 EJDE-207/75 SOME PROPERTIES OF MEROMORPHIC SOLUTIONS 5 If k and a k 0 is a constant. Then, we have from 3.2 that P q z, 0 = [pqz + p z q ]pz az + b cpz = qk + q k a2 kz 2k Since q, we have q k + q k 2.2 that Then, we obtain N r, fz pz = N r, 0, then P q z, 0 0. Thus, we have by Lemma mr, = Sr, g. g Hence, it follows that f p = σf. If a = 0 and pz = β E, then we have = T r, g + Sr, g = T r, f + Sr, f. 3.4 gz P q z, 0 = 2β 2 cβ b 0. Set gz = fz β, by using the same argument as above, we can obtain f β = σf. Therefore, we can obtain that the Borel exceptional values of fz can only come from the set E = {z 2z 2 cz b = 0}. ii From.5, we have It follows from Lemma 2. that By applying Lemma 2.4 for.5, we have And by Lemma 2.3 we obtain fz[fqz + f z ] = az + b + cfz. 3.5 q m r, fqz + f z q = Sr, f. 3.6 T r, fqz + f z q = T r, f + Sr, f. 3.7 N r, fqz + f z q Nr, fqz + N r, f z q = 2 + onr, f 3.8 on a set of lower logarithmic density. Thus, combining 3.6 and 3.7, we have Hence, we have T r, f 2 + onr, f + Sr, f. σfz. 3.9 fz Next, we prove that qfz fz. Set z = qw, then we can rewrite.5 as the form fq 2 w + fw = aqw + b + c. 3.0 fqw Then it follows that fqw[fq 2 w + fw] = aqw + b + cfqw. 3. Since q fw = fqw fw, we have fqw = q fw + fw and fq 2 w = q fqw + q fw + fw. Substituting them into 3., we obtain [ q fw + fw][ q fqw + q fw + 2fw] = aqw + b + c[ q fw + fw],

6 6 H. Y. XU, S. Y. LIU, X. M. ZHENG EJDE-207/75 i.e., 2fw 2 =[ q fqw + 3 q fw c]fw aqw + b + [ q fqw + q fw c] q fw. 3.2 Since fz is a zero order transcendental meromorphic function and z = qw, by Lemma 2.3, we obtain that fw is of zero order. Thus, by Lemma 2.3 again, we have that q fw, q fqw are of zero order. Set 2 qfw := q q fw, so we have q fqw = 2 qfw + q fw. Since q fw is of zero order, and by Lemma 2.3 we have It follows that Nr, 2 qfw 2Nr, q fw + Sr, f. 3.3 Nr, q fqw 3Nr, q fw + Sr, f. 3.4 Thus, from 9 and 3.4 we have 2Nr, fw =N r, [ q fqw + 3 q fw c]fw aqw + b + [ q fqw + q fw c] q fw That is, Then, it follows that Nr, fw + 9Nr, q fw + Olog r + Sr, f. Nr, fw 9Nr, q fw + Sr, f. 3.5 q fw. 3.6 fw Since fw is of zero order, by Lemma 2.3 and z = qw we have = q fw q f z q =, q fz = fw f z q =. fz Hence, q fz From this inequality and 3.9 we have q fz. 3.7 fz σfz. 3.8 fz So, we have T r, q fz 2T r, fz + Sr, f by Lemma 2.3; that is, σfz σ q fz. Thus, combining this and 3.8, we have = = σ q fz = σfz. fz q fz The proof of Theorem.3 is complete.

7 EJDE-207/75 SOME PROPERTIES OF MEROMORPHIC SOLUTIONS 7 4. Proof of Theorem.4 Suppose that fz is a zero order transcendental meromorphic solution of.6. We will consider the following two cases. i a 0. If pz is a polynomial of degree k and pz = a k z k Let g z = fz pz. Substituting fz = g z + pz into equation.6, we have It follows that g qz + pqz + g z q + pz q = az + b[g z + pz] + c [g z + pz] 2. P q z, g :=[g qz + pqz + g z q + pz q ][g z + pz] 2 From this equality, we have + az + b[g z + pz] + c [g qz + pqz + g z q + pz q ] = P q z, 0 = [pqz + p z q ]pz2 + az + bpz + c pqz p z. 4.2 q If k = 0 and a 0 α C \ {0}, then P q z, 0 = 2α 3 + αaz + b + c 2α 0. If k and a k 0 is a constant. Then, we have from 4. that P q z, 0 = [pqz + p z q ]pz2 + az + bpz + c pz p z q = q k + q k a 3 k z 3k Since q, we have q k + q k 2.2 that Then, we obtain N r, fz pz = N r, 0, then P q z, 0 0. Thus, we have by Lemma mr, g = Sr, g. It follows hat f p = σf. If a = 0 and pz = β E, then we have = T r, g + Sr, g = T r, f + Sr, f. g z P q z, 0 = 2β 3 + b 2β + c 0. Set g z = fz β, by using the same argument as above, we can obtain f β = σf. Therefore, we can obtain that the Borel exceptional values of fz can only come from the set E = {z 2z 3 + b 2z + c = 0}. If c 0, then we have from.6 that P q z, f := fz 2 [fqz + f z q ] + az + bfz + c fqz fz q. Hence, we obtain P q z, 0 c 0. Using a similar method as above, we obtain f = σf. ii From.6, we have fz 2 [fqz + f z q ] = fqz + fz az + bfz c. 4.3 q

8 8 H. Y. XU, S. Y. LIU, X. M. ZHENG EJDE-207/75 It follows from 3.5 that m r, fqz + f z q = Sr, f. 4.4 By applying Lemma 2.4 for.6, we have T r, fqz + f z q = 2T r, f + Sr, f. 4.5 By Lemma 2.3 we obtain N r, fqz + f z q Nr, fqz + N r, f z q = 2 + onr, f 4.6 on a set of lower logarithmic density. Thus, combining 3 and 32, we have Hence, Next, we prove that.6 in the form Then it follows that T r, f 2 + onr, f + Sr, f. qfz σfz fz fq 2 w + fw = fz Set z = qw, then we can rewrite aqw + bfqw + c fqw fqw 2 [fq 2 w + fw] = fq 2 w + fw aqw + bfqw c. 4.9 Since q fw = fqw fw, we have fqw = q fw + fw and fq 2 w = q fqw + q fw + fw. Substituting these two equalities in 4.9, we obtain Thus, we obtain where [ q fw + fw] 2 [ q fqw + q fw + 2fw] = q fqw + q fw + 2fw aqw + b[ q fw + fw] c, 2fw 3 = Awfw + Bw q fw + Cw, 4.0 Aw =[ q fqw + 5 q fw]fw + 4 q fw q fw q fqw + aqw + b 2, Bw = q fqw q fw + q fw 2 + aqw + b, Cw = c q fqw. Thus, by Lemma 2.3 and from 3.4 we have That is, 3Nr, fw =Nr, Awfw + Bw q fw + Cw Then, it follows from 37 that q fw 2Nr, fw + 9Nr, q fw + Olog r + Sr, f. Nr, fw 9Nr, q fw + Sr, f fw

9 EJDE-207/75 SOME PROPERTIES OF MEROMORPHIC SOLUTIONS 9 Since fw is of zero order, by Lemma 2.3 and z = qw we have = q fw q f z q =, q fz = fw f z q =. fz Hence, we obtain q fz. 4.3 fz Thus, from this inequality and 4.7 we have σfz. 4.4 q fz fz Then we have T r, q fz 2T r, fz+sr, f by Lemma 2.3; that is, σfz σ q fz. Thus, combining this and 4.4, we have = = σ q fz = σfz. fz q fz The proof of Theorem.4 is complete. 5. Proof of Theorem.5 For the convenience of the reader, we use the notation form the proof of Theorem.3i. Suppose that fz is a zero order transcendental meromorphic solution of.7. We will consider the following two cases. i Let a = 0 and pz = β E. Using the same methods as in the proof of Theorem.3i, we have P q z, 0 = 2β 3 bβ c 0. Thus, f β = σf. Hence, the Borel exceptional values of fz can only come from the set E = {z 2z 3 bz c = 0}. ii From.7, we have fz 2 [fqz + f z ] = az + bfz + c. 5. q It follows from Lemma 2. that m r, fqz + f z q = Sr, f. 5.2 By Lemma 2.3 we obtain N r, fqz + f z q Nr, fqz + N r, f z q = 2 + onr, f 5.3 on a set of lower logarithmic density. If c 0, by applying Lemma 2.4 for.7, we have T r, fqz + f z q = 2T r, f + Sr, f. 5.4 Thus, it follows from that T r, f + onr, f + Sr, f. 5.5

10 0 H. Y. XU, S. Y. LIU, X. M. ZHENG EJDE-207/75 If c = 0, by applying Lemma 2.4 for.7 and since a + b + c = a + b 0, we have T r, fqz + f z q = T r, f + Sr, f. 5.6 Thus, it follows from 5.2, 5.3 and 5.6 that From 5.5 and 5.7 we have Next, we prove that T r, f 2 + onr, f + Sr, f. 5.7 qfz σfz argument as in Theorem.4ii, we have where fz fz Set z = qw, by using the same 2fw 3 = Awfw + Bw q fw + Cw, 5.9 Aw = [ q fqw + 5 q fw]fw + 4 q fw q fw q fqw aqw b, Bw = [ q fqw + q fw] q fw aqw b, Cw = c. Thus, by Lemma 2.3 and from 3.4 we have That is, 3Nr, fw =Nr, Awfw + Bw q fw + Cw Then, it follows that 2Nr, fw + 5Nr, q fw + Olog r + Sr, f. Nr, fw 5Nr, q fw + Sr, f. 5.0 q fw. 5. fw Since fw is of zero order, by Lemma 2.3 and z = qw we have = q fw q f z q =, q fz = fw f z q =. fz Hence, we obtain Thus, by 5.8 we have q fz q fz. 5.2 fz σfz. 5.3 fz So, we have T r, q fz 2T r, fz + Sr, f by Lemma 2.3; that is, σfz σ q fz. Thus, combining this and 5.3, we have = = σ q fz = σfz. fz q fz The proof of Theorem.5 is complete.

11 EJDE-207/75 SOME PROPERTIES OF MEROMORPHIC SOLUTIONS Acknowledgments. This work was supported by the National Natural Science Foundation of China 56033, , 30233, , the Natural Science Foundation of Jiangxi Province in China 205BAB20008, 207BAB20002, the Natural Science Foundation of Shaanxi Province in China 207JM00, and the Foundation of Education Department of Jiangxi GJJ6094,GJJ50902 of China. The authors want to thank the anonymous referees for reading the manuscript very carefully and making a number of valuable comments which improved this article. References [] D. C. Barnett, R. G. Halburd, R. J. Korhonen, W. Morgan; Nevanlinna theory for the q-difference operator and meromorphic solutions of q-difference equations, Proc. Roy. Soc. Edin. Sect. A Math., , [2] Z. X. Chen; On properties of meromorphic solutions for some difference equations, Kodai Math. I., 34 20, [3] Z. X. Chen, K. H. Shon; Properties of differences of meromorphic functions, Czechoslovak Math. J., , [4] Z. X. Chen; Value distribution of meromorphic solutions of certain difference Painlevé equations, J. Math. Anal. Appl., , [5] Y. M. Chiang, S. J. Feng; On the Nevanlinna characteristic of fz + η and difference equations in the complex plane, Ramanujan J., , [6] A. S. Fokas; From continuous to discrete Painlevé equations, J. Math. Anal. Appl., , [7] L. Y. Gao, M. L. Wang; The transcendental meromorphic solutions of composite functional equations, J. Jiangxi Norm. Univ. Nat. Sci., , [8] B. Grammaticos, F. W. Nijhoff, A. Ramani; Discrete Painlevé equations, The Painlevé property, CRM Ser. Math. Phys., Springer, New York, 999, pp [9] G. G. Gundersen, J. Heittokangas, I. Laine, J. Rieppo, D. Q. Yang; Meromorphic solutions of generalized Schröder equations, Aequationes Math , [0] R. G. Halburd, R. J. Korhonen; Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl., , [] R. G. Halburd, R. J. Korhonen; Finite order solutions and the discrete Painlevé equations, Proc. London Math. Soc , [2] R. G. Halburd, R. J. Korhonen; Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math., , [3] W. K. Hayman; Meromorphic Functions, The Clarendon Press, Oxford, 964. [4] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, K. Tohge; Complex difference equations of Malmquist type, Comput. Methods Funct. Theory, 200, [5] I. Laine; Nevanlinna theory and complex differential equations, Walter de Gruyter, Berlin, 993. [6] I. Laine, C. C. Yang; Clunie theorems for difference and q-difference polynomials, J. London Math. Soc., , [7] I. Laine, C. C. Yang; Value distribution of difference polynomials, Proc. Japan Acad. Ser. A, , [8] G. Li, M. Chen; Infinite horizon linear quadratic optimal control for stochastic difference time-delay systems, Adv. Difference Equations, , Art. 4. [9] L. W. Liao; The new developments in the research of nonlinear complex differential equations, J Jiangxi Norm. Univ. Nat. Sci., , [20] L. W. Liao, C. C. Yang; Some new and old unsolved problems and conjectures on factorization theory, dynamics and functional equations of meromorphic functions, J Jiangxi Norm. Univ. Nat. Sci , [2] P. Painlevé; Mémoire sur les équations différentielles dont l intégrale générale est uniforme, Bull. Soc. Math. France, , [22] C. W. Peng, Z. X. Chen; On properties of meromorphic solutions for difference Painlevé equations, Advances in Difference Equation, , no. 23, pp. -5.

12 2 H. Y. XU, S. Y. LIU, X. M. ZHENG EJDE-207/75 [23] X. G. Qi, L. Z. Yang; Properties of meromorphic solutions of q-difference equations, Electronic Journal of Differential Equations, , No. 59, pp. -9. [24] J. Wang, K. Xia, F. Long; The poles of meromorphic solutions of Fermat type differentialdifference equations, J Jiangxi Norm. Univ. Nat. Sci., , [25] L. Yang; Value distribution theory, Springer-Verlag. Berlin, 993. [26] H. X. Yi, C. C. Yang; Uniqueness theory of meromorphic functions, Kluwer Academic Publishers, Dordrecht, 2003; Chinese original: Science Press, Beijing, 995. [27] J. L. Zhang, R. Korhonen; On the Nevanlinna characteristic of fqz and its applications, J. Math. Anal. Appl., , [28] X. M. Zheng and Z. X. Chen; Some properties of meromorphic solutions of q-difference equations, J. Math. Anal. Appl., , [29] X. M. Zheng, Z. X. Chen; On properties of q-difference equations, Acta Mathematica Scientia, 32B 2 202, Hong Yan Xu corresponding author School of mathematics and statistics, Xidian University, Xi an, Shaanxi 7026, China. Department of Informatics and Engineering, Jingdezhen Ceramic Institute Xiang Hu Xiao Qu, Jingdezhen, Jiangxi , China address: xhyhhh@26.com San Yang Liu School of mathematics and statistics, Xidian University, Xi an, Shaanxi 7026, China address: liusanyang@26.com Xiu Min Zheng Department of Mathematics, Jiangxi Normal University, Nanchan, Jiangxi , China address: zhengxiumin2008@sina.com